In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?
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-LD
________________________________________
my faith
In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?
Ah, now that is a core question. It may take more than a bit of explaining though .... and beware there is no neat/pat answer that will likely satisfy your intuition. It really is quite a paradigm shift. Go and get a cup of your favorite brew before reading. I'll give you Mike's Tour Of GR! :-)
Firstly : one has to include time in the geometry. Just talking about 3-D space alone won't cut it. Try this simple example. Fire a cannonball at some fixed angle into the air. You'll find that where it lands depends ( ignoring lots of other fiddly non-gravitational stuff ) on the velocity with which it was fired. Indeed for given initial and final points even, there are two trajectories - the high lob and the low, flat shot - if you allow both the angle and direction of velocity to change.
The same applies to orbits around a central body, like planets and moons, where the subsequent evolution of the path requires considering the magnitude as well as the direction of the velocity vector. One could imagine the Earth in it's current orbital position around the Sun, with it's usual velocity, and it'll keep circling. But suppose it was at exactly the same position but with say, five times it's usual speed - it certainly won't hang around the Sun for much longer. So there's a time ( rate of change ) required in the explanation. That's as true for GR as it is for classical explanations.
Secondly : when we say geometry it's crucial, though no doubt annoying, to have to consider the basis of measurement. As Einstein found out with Special Relativity, some 'obvious' ideas in classical physics turned out to be wrong, or at least misleading and approximate. So there are many choices of geometric description, and for Einstein the challenge was to come up with a formulation that would be physically true ( same predictions ) regardless of special choices of how/where/when the geometry is defined. Thus the Earth ought continue to orbit the Sun, say, as viewed by a whole range of observers in various positions, with different clocks and measuring sticks.
Fortunately much of this was already done, incidentally, by a chap called Riemann. He studied what geometry would be like if it wasn't according to Euclid/Pythagorus etc. His essential breakthrough was a way of describing geometry locally and within the thing being discussed. So one might look at an apple and say : it is round much like a sphere, more so near the top where the stem is, is dimpled/puckered either end and is pretty smooth overall. These are words which are really 'external' descriptors. Riemann's way of deducing this equivalently was to say : I have a point on the apple and I compare two paths diverging from that point, further down each path I find they meet each other again. He comes up with a 'value' at each point on the apple, such that if one considers the totality of all these values you could arrive at what it would look like if you did see the apple from afar.
Now the reasons for going to a local ( in time as well as space ) rather than a general description is several fold :
- 3D in space plus one in time is 4 dimensions. Hard to visualise per se.
- to split the problem into two parts. Begin by stating the geometry in terms of what distribution of matter/energy produces it, then given that, see what response some object has in that geometry. Hence 'matter tells space how to warp, and space tells matter how to move'.
- spacetime is flat whenever viewed from close enough. This means that for a short enough time and/or for a small enough distance any movement looks Euclidean ( or Gallilean or Newtonian ).
So thirdly : how do you describe movement in detail in this framework? Well those 'values' I mentioned above are really a set of values at each point in spacetime. They 'explain' how you transition from one point to the next if you are freely falling ( only subject to gravity ). This is where the 'warping' business comes in : at each point in spacetime where gravity is acting ( and gravity is everywhere acting on everything ), these 'metric tensors' are a local guide to how directions change and thus which way to go next. I've described in another thread an analogy with small villages on some undulating landscape. At the centre of each hamlet is a signpost where roads intersect that has directions indicating the way to nearby villages. The metric prescribes how these signposts ought vary from place to place. [ The full horror is calculus, infinitesimals and equations with partial derivatives ... ]
If you were describing a globally flat spacetime ( unrealistically meaning no mass or energy was about ) then the metrics will state that the signposts don't vary from village to village. If you grind through the math in this scenario then you'd wind up describing the law of inertia in free space, where things just keep going if they are already. Specifically they wont deviate from a straight line.
Fourthly : I have left light out thus far. In GR the phrase 'straight line' is replaced by 'the path that light follows' or 'null geodesic'. For ordinary life these are easily seen to be the same thing. If you can arrange matters to view three objects by eye and see that they overlap/occult one another simultaneously then we say they are in a line. Bricklayers, surveyors, shooters etc explicitly do this all the time. So one way of mapping the curvature of spacetime is to study what the light rays are up to. The time component comes out as a change in the frequency of the light radiation, and thus is a measure of how time ( ie. clocks ) varies around and about.
Add in that any non-zero rest mass can't match ( or exceed ) light speed then you have an overall rule that matter won't escape the confines of the light paths. Hence the 'cone' analogy to spacetime points, where curvature means the cones are wobbly shaped if compared to the flat case. The way the cones change their shape from here to there is encoded in the metric.
So finally back to Newton's case of a single central mass that influences another one some distance away - your originally query. The good & bad news is that we have an approximate but not exact solution to the GR equations for this. It's still pretty good and has been observationally well confirmed more than a few times though ( eclipses, Mercury .. ). It was figured out by an artillery officer on the eastern European front in WWI, and he died not long after mailing it to Einstein. It's the ( Karl ) Schwartzchild ( maybe without the 't' ? ) solution, and has been expanded upon by others to include rotation and electric charge too ( see Kerr ).
The solution has a special quality with regard to a certain distance from the central body, the Schwartzchild radius. This radius depends on the mass of the body and some fundamental universal constants. If a body happens to lie completely within it's own Schwartzchild radius it will become a black hole. I won't recount all the observational features of black holes bar the prime one - that not even light can travel from within that radius to without ( discounting quantum effects ). For me that radius is about that of a proton I think, for the Earth about an apple size, for the Sun about the width of a mountain. As Mike/Earth/Sun each are not compact or dense enough - insufficient mass within a given volume - then while not black holes, there may be some measurable deviation of light rays passing by ( as viewed from far away ). An eclipse just after WW1 discovered the effect for the Sun, later technology in the 1960's ( plus the GPS more recently ) confirmed that for the Earth. No one has yet come along to demonstrate the effect around me though .....
Personally I try to avoid the word 'curvature' or at least mentally substitute it with the phrase 'observers differ'. That way time can be 'curved' by differently situated clocks progressively disagreeing with each other.
For the mathematically inclined : deep in the gore of GR is a 4 by 4 matrix that is used to convert/connect one spacetime vector/point/event to another. There is a row for each of the space directions plus one for time, and each of those rows has four columns - for each space direction plus one for time. This is one way of representing the metric. The metric is one thing you definitely want to discover for a given problem. If you know it then you can say how things will 'fall' or behave in the absence of non-gravitational forces.....
Cheers, Mike.
( edit ) To be more precise, I ought say by 'metric tensors' I mean a metric tensor which is evaluated at many points. A tensor is sort of a multi-functional function. So instead of a single valued function - one number in, one number out - a tensor can have many things both in and out. In a sense we bundle lots of single-valued functions together, for instance how stuff in the z-direction depends on time, or how time depends on stuff in the x-direction. But they work as a group, and reasonable ideas of symmetry ( ie. reasonable universes without surprising behaviours that we haven't yet seen ) contract 4 x 4 = 16 functions to 10 independent ones.
Another way to visualise is : at each point in spacetime ( each moment in space & for each instant ) you have this associated tensor 'gadget' or 'box' that you can crank. We don't of course have an infinite listing of boxes, but Einstein's equation that governs their character. What remains is initial/starting/boundary conditions. We may well know how things vary from ( spacetime ) point to point but that still leaves some freedom in choice of the 'baseline'. Alot of discussions I read about GR astronomical problems divulge alot of assumptions upon these conditions. In a way one might solve say two neutron stars circling each other, however they aren't really alone so you have to 'connect' their behaviour to the rest of the universe at the 'boundaries'. Boundary also applies to the time co-ordinate, thus from whence and until whence is quite relevant.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
That's how I see it. Take a four-dimensional Riemannian manifold and endow it with a pseudoeuclidean metric (that of special relativity). Then calculate the tangent space at a chosen point. You have to differentiate the manifold using differential operators. Question: do they form a Lie algebra? If yes, which one? I have been unable to answer this question. But is it a good question?
Tullio
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It's the ( Karl ) Schwartzchild ( maybe without the 't' ? ) solution...
Indeed it's Schwarzschild (black shield, not child ;-)
Gruß,
Gundolf
It's the ( Karl ) Schwartzchild ( maybe without the 't' ? ) solution...
Indeed it's Schwarzschild (black shield, not child ;-)
Ah, so not 'the child of Schwartz' then. Thanks, I've seen many spellings. With the 't' is probably an anglicized mangling ... :-)
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
That's how I see it.
I'm relieved! It's a toughy to understand, much less explain! :-)
Take a four-dimensional Riemannian manifold and endow it with a pseudoeuclidean metric (that of special relativity). Then calculate the tangent space at a chosen point. You have to differentiate the manifold using differential operators. Question: do they form a Lie algebra? If yes, which one? I have been unable to answer this question. But is it a good question?
Don't know much about Lie algebras per se, except that they mean 'smooth', 'differentiable', 'continuous' and what not. So that's tantamount to asking if we can ( or not ) quantize spacetime? Good question indeed ...
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?
Ah, now that is a core question. It may take more than a bit of explaining though .... and beware there is no neat/pat answer that will likely satisfy your intuition. It really is quite a paradigm shift. Go and get a cup of your favorite brew before reading. I'll give you Mike's Tour Of GR! :-)
Firstly : ...
Secondly : ...
So thirdly : ...
Fourthly : ...
So finally ...
Thanks for the PG Tips.
(http://www.youtube.com/watch?v=gfG2ZujlIZU&feature=related)
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-LD
________________________________________
my faith
I originally asked, "In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?" To which Mike replied (the quotes) and then my replies below his replies. (I edited the 'hec' out of the quotes and lost the original flow. So now I have to explain the above... like a pendulum do.)
...
These are words which are really 'external' descriptors. Riemann's way of deducing this equivalently was to say : I have a point on the apple and I compare two paths diverging from that point, further down each path I find they meet each other again. He comes up with a 'value' at each point on the apple, such that if one considers the totality of all these values you could arrive at what it would look like if you did see the apple from afar.
May I have some of the math? I have an MS in Physics, and have taken a GR course (millennium ago), and am now reading the Princeton Phone book (MTW's blerb about GR).
E.g. are you refering to the affine connection? [tex]\Gamma^{\alpha}_{\mu\nu}[/tex] the metric tensor?? [tex]g_{\mu\nu}[/tex] BOTH???
"Why no tex? ... you have no tex!?! AHHH!! HE HAS NO TEX!!!" (apologies to Mike Judge)
...
- to split the problem into two parts. Begin by stating the geometry in terms of what distribution of matter/energy produces it,
[tex]T_{\mu\nu}[/tex]?
...
then given that, see what response some object has in that geometry. Hence 'matter tells space how to warp, and space tells matter how to move'.
OHhhh... I see. Since the mass moves in a way according to the spacetime!
...
- spacetime is flat whenever viewed from close enough. This means that for a short enough time and/or for a small enough distance any movement looks Euclidean ( or Gallilean or Newtonian ).
Yes, b/c measurements have a certain precision. "Flat" really means "I cannot measure curvature below the precision of my equipment."
...
So thirdly : how do you describe movement in detail in this framework? Well those 'values' I mentioned above are really a set of values at each point in spacetime. They 'explain' how you transition from one point to the next if you are freely falling ( only subject to gravity ). This is where the 'warping' business comes in : at each point in spacetime where gravity is acting ( and gravity is everywhere acting on everything ), these 'metric tensors' are a local guide to how directions change and thus which way to go next.
Yeah... so the mass distribution (typically a sphere, ala Scwarzschild) determines [tex]g_{\mu\nu}[/tex] and then that determines the proper interval. [tex]g_{\mu\nu}[/tex] tells things how two points are connected - "curved" via [tex]g_{\mu\nu}[/tex] or "flat" via [tex]\eta_{\mu\nu}[/tex].
Right, in short: [tex]ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}[/tex]
...
Fourthly : I have left light out thus far. In GR the phrase 'straight line' is replaced by 'the path that light follows' or 'null geodesic'.
Is this the role of the Killing vector? If one solves for X in the Killing equation, then one knows the Killing vector, which is the path of a photon in this metric (with given mass distribution), which is called the "geodesic". IOW, solve for X in:
[tex]X_{\mu};_{\nu} + X_{\nu};_{\mu} = 0[/tex]
...
For ordinary life these are easily seen to be the same thing. If you can arrange matters to view three objects by eye and see that they overlap/occult one another simultaneously then we say they are in a line.
Right... we say they are in a "straight" line, but in reality they are following along the geodesic! Yes?? :)
...
The time component comes out as a change in the frequency of the light radiation, and thus is a measure of how time (i.e. clocks) varies around and about.
Ohhh, so that's how we know the spacetime is curved - by the change in the color of the light as it travels from one point to another. Yes?? :(
Yes, on the light cone analogy. I think I am understanding it now.
...
we have an approximate but not exact solution to the GR equations for this.
I thought it was exact. Was this the problem with the cosmological term?
...
It was figured out by an artillery officer on the eastern European front in WWI, and he died not long after mailing it to Einstein. It's the ( Karl ) Schwartzchild (maybe without the 't' ?) solution,
Oh wow... I did not know that. What a shame.
...
Personally I try to avoid the word 'curvature' or at least mentally substitute it with the phrase 'observers differ'. That way time can be 'curved' by differently situated clocks progressively disagreeing with each other.
Well, if my replies are correct, then I can accept curvature since it is really how two points are connected - "geodesic-ally" - via the [tex]g_{\mu\nu}[/tex]. Yes?
Thanks Mike!
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-LD
________________________________________
my faith
May I have some of the math? I have an MS in Physics, and have taken a GR course (millennium ago), and am now reading the Princeton Phone book (MTW's blerb about GR).
Hadn't heard that ( phone book ) phrase, but I see what you mean! :-)
I've been keeping the description away from math specifics because (a) I'm not that knowledgeable enough to frame it correctly ( but I'm studying .... ) and (b) tensor arithmetic ( gymnastics with indices ) tends to obscure the physical meaning.
OHhhh... I see. Since the mass moves in a way according to the spacetime!
The underlying model is a smooth manifold so that it can be differentiated as many times as needed, and is classically/continuously so ( no weird quantum/foamy bits at really small scales ).
Yes, b/c measurements have a certain precision. "Flat" really means "I cannot measure curvature below the precision of my equipment."
Yup indeed, but I really should have said "Minkowskian" ( as per Special Relativity ) so that
ds^2 = - dt^s + dx^2 + dy^2 + dz^2
becomes the infinitesimal line element ( there are several conventions possible here ). Or that metric tensor becomes :
with all the "cross" partial derivatives being zero ie. not the case with more general curvature. You can draw simple spacetime diagrams so that ( say, time on vertical axis and a space dimension on the horizontal ) is easily analysed with 'flat' triangles and whatnot. Meaning that it is sufficient to know the co-ordinate differences of events to yield the lengths of non-infinitesimal separations. With GR the co-ordinate differences don't give that, you have to "integrate along" as the metric changes from point to point ( a landscape stuffed full of an infinite number of villages lying along your curve, each an infinitesimal jump away from the previous and the next ). The co-ordinate values of two particular events of interest are endpoints to that integration ( I start and end at some named villages ).
Yeah... so the mass distribution (typically a sphere, ala Scwarzschild) determines [tex]g_{\mu\nu}[/tex] and then that determines the proper interval. [tex]g_{\mu\nu}[/tex] tells things how two points are connected - "curved" via [tex]g_{\mu\nu}[/tex] or "flat" via [tex]\eta_{\mu\nu}[/tex].
Right, in short: [tex]ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}[/tex]
Yup, eta is the straight co-ordinate difference approach whereas gee is a function ( group thereof ..... ) that varies across your spacetime landscape.
Is this the role of the Killing vector? If one solves for X in the Killing equation, then one knows the Killing vector, which is the path of a photon in this metric (with given mass distribution), which is called the "geodesic". IOW, solve for X in:
[tex]X_{\mu};_{\nu} + X_{\nu};_{\mu} = 0[/tex]
Dunno. Probably :-)
All I know of that is the Killing vector tells you where to go to preserve/not-distort distances on an object.
Right... we say they are in a "straight" line, but in reality they are following along the geodesic! Yes?? :)
Well, we say a straight line is the geodesic. By fiat. Fait a compli. Say it is so as an axiom, then move on with deductions assuming that. In a sense it is word-play, but I think there is a variational principle here : one has two endpoints ( spacetime events ) such that over all possible paths between, the one with the minimum "total distance" is that which light takes ( all 'adjacent' paths are longer ). You can't beat it.
Ohhh, so that's how we know the spacetime is curved - by the change in the color of the light as it travels from one point to another. Yes?? :(
Well anything time-dependent, but as photons get involved sooner or later in any practical time definition then that's a good indicator of change.
There's a neat short section on page 26-27 of MTW that compares observers with 'good' and 'bad' clocks : essentially one can make acceleration appear/disappear solely by choice of clock behaviour ( position measurements unchanged ). Reversing the approach, one can say that by choosing a local spacetime metric as flat ( inertial/unaccelerated ) forces a re-definition of the time standard ie. clocks alter. So things become inertial locally by suitable choice of clock. Which clock? The one that makes acceleration go away!! :-)
Yes, on the light cone analogy. I think I am understanding it now.
Like a Hogwart's hat, it's roughly conical but squashy. :-)
I thought it was exact. Was this the problem with the cosmological term?
Nope, it's linear approximations on non-linear equations. That's the whole rub of GR. Essentially the field itself has energy, so that 'feeds back'. A true & exact solution must encompass it's own presence, so to speak. QCD with quarks and gluons has a similiar character. So the mass of a proton say, can be partitioned into 'bare' quark masses plus a mass due to interaction energies of the gluons. Likewise a binary neutron star system has alot of 'gravitational self energy' meaning the entire system energy is well more than the total of the separated masses ( say around 30% - ish more ?? ).
Oh wow... I did not know that. What a shame.
Poor lad, got a crappy skin infection from the horrible mud of the trenches and died from blood poisoning ( no antibiotics then ).
Well, if my replies are correct, then I can accept curvature since it is really how two points are connected - "geodesic-ally" - via the [tex]g_{\mu\nu}[/tex]. Yes?
Yup, what is straight for one guy with his rulers/clocks is wiggly for another with different ones. This is more than SR, where for a given relative speed you can stroll around one or other frame with a metric that is constant over the entire given frame. In GR you have to live with rulers and clocks that morph even in the same frame. So as you stroll from village to village, clock in one hand and ruler in the other, they are subtly changing as you go .... remember the metric is an agglomeration of functions relating your space and time degrees of freedom, the specific evaluation of which is time/space dependent.
Thanks Mike!
A pleasure, but beware I could be easily lacking the proper rigour here. Thanks for the W&G links too, I didn't know they did TV ads !? :-)
Cheers, Mike.
[ edit ] You could of course integrate along a path in Minkowski/SR frames, but you get a simple answer - the same as found by subtracting co-ordinates and then doing a spot of ( flat ) trigonometry.
[ edit ] Another aspect is somewhat more philosophical, but it works. Because we have yet to find any phenomena that are not subject to gravitational or inertial effects ( NB one may have to look hard though ) we say gravity/inertia is universal. This is tantamount to saying that gravity/inertia is not a property of each specific body/particle so much as a characteristic of the spacetime they exist in. We attribute 'free fall' ( no non-gravitational accelerations ) to the background and not to the particles per se. The gravitational force disappears. So in a way that is a neat compression of the thinking/algorithm, find out what spacetime is up to and then given that : all bodies will behave similiarly. { Except one trouble, the presence of a body will change the solution and the bigger the mass/energy the more the change }
[ edit ] Note with regard to the 'good' and 'bad' clocks. It's the second derivative ( of one clock reading compared to the other ) which has to be non-zero. So we're not talking of the standard SR time dilation ( constant ratio between clocks in different frames ), but that said dilatation changes as time proceeds .... this is how an astronaut waving goodbye as he/she descends towards a black hole event horizon gradually slows down and 'freezes' as viewed by a distant observer. Black holes had been called 'frozen stars' prior to Penrose et al in the 1960's . Black holes are black because the light is infinitely frequency shifted to zero frequency, or equivalently the energy barrier to surmount is always greater than what any photon can start off with ( an infinite shift beats any finite frequency ).
[ edit ] Is it 'dilation' or 'dilatation' ?? I'm never quite sure of that one !! :-)
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
A concrete but very artificial 'example' as regards the time change. Strictly speaking this likely falls over as I'm attributing all changes to the time axis alone ... so this is an 'in principle' explanation for 'some universe'. :-)
Suppose I have an 'ordinary' clock marking intervals for a falling body. Say we're on the Moon, so as to exclude all that air resistance stuff. So we have 'free fall' or no non-gravitational forces. Then if I look at the distance fallen for t = 0, 1, 2, 3, 4, 5 .... seconds I'll get ( distance ) x = 0, 1, 4, 9, 16, 25 ..... units. That is : x goes like t^2.
Now let's have another clock, but fudged so that it marks time but according to the square root of the reading on the first clock ( T = t^(1/2) ) . So on this clock, at time T = 0, 1, 2, 3, 4, 5 .... seconds I'll get ( distance ) x = 0, 1, 2, 3, 4, 5 .... units !!! So : x goes like the square root of [ the square of time ] ...
[ I haven't changed the length ruler ]
In the first case my distance is quadratic with time, whereas in the second it is linear. The first case says I have an acceleration ( dx^2/dt^2 != 0 ). Is dT^2/dt^2 non-zero? You bet it is! dT^2/dt^2 = (-1/4) * t^(-3/2). And dx^2/dT^ 2 = 0, so I have un-accelerated behaviour by that choice of 'dodgy' clock.
Note that the longer you run things, the seconds of 'T' time represent shorter intervals in 't' time. So any physical process is doing less per equal 'T' interval compared with the 't' interval. Or put another way : to make the 'T' time system mark those distance increments as equal every 'T' second, thus eliminating the acceleration in the 'T' frame, then for every 't' second ( with the speed increasing in the 't' system ) I have to jump in quicker on each 't' tick.
Cheers, Mike.
[ edit ] So for when the first clock 'strikes' t = 0, 1, ~1.414, ~1.732, 2, ~ 2.236, ~2.449, ~2.646, ~2.828, 3 ...... the second clock 'strikes' T = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .... hence t(T+1) - t(T) = 1, ~0.414, ~0.318, ~0.268, ~0.236, ~0.213, ~0.197, ~0.182, ~0.172 ...
[ edit ] This is, of course, a local ( in time as well as space ) effect/argument. I'll hit something eventually as the body falls and/or the acceleration increases as I get closer to whatever is the central gravitating body. Generally non-gravitational forces interrupt our pure GR discussion by eliminating the 'free fall' assumption. Good thing too ... else what would stop the mass/energy density rising to make black holes far more common?? It's quantum that stops opposite electric charges sitting on top of one another. ;-) :-)
[ edit ] So if I could instantly flip my metabolism, thinking etc over to the 'T' rate from the 't' then : the acceleration would go away, and other stuff happening around the place would progressively slow down by my perception. If you have a functional dependence b/w 'T' and 't' other than the above ( parabola on it's side ) the arguments still qualitatively hold. Question : could the strengthening of a gravitational field speed up time ???? If not, why not ??? What ( pretty fundamental ) change would be needed for that to occur ???? :-) :-)
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Question : could the strengthening of a gravitational field speed up time ???? If not, why not ??? What ( pretty fundamental ) change would be needed for that to occur ???? :-) :-)
By 'speeding up' in the context as above means that for each 'T' second passing by, more than one second passes in 't' time. So instead of t(T+1) - t(T) going from unity down to zero, it goes upwards instead. That way from a 'T' time aligned persona more is physically happening per 'T' second. But I still want to get rid of any accelerations by using the 'T' clock ....
The only way ( by time adjustment alone ) I can get the same distance covered during longer 't' intervals ( ie. removing acceleration ) is to have the distance travelled per 't' second decrease. So the further the object falls the slower it goes. There are no non-gravitational forces acting. And I'm getting closer towards a central body.
So now I'm falling 'down' and going slower with time. Hmmmm .... we would call this anti-gravity, would we not?? :-) :-)
All this is tantamount to saying that with our 'usual' assumptions about the universe no clock will run faster than one which is clear of all gravitational fields. Any field increase must slow the clock ( go closer to some mass ). You can speed a clock up but only by going from a stronger field to a weaker one - like coming back up from nearby a black hole event horizon to some distant position.
[ Another way around is to find out about the position of objects earlier than otherwise. Supraluminal signals will cause us to record any distance increments 'sooner'. By that I mean the case of normal attractive gravity but with tachyons now. Essentially this is what Newton assumed, instantaneous transmission, an infinite light speed and no clock variations of any sort. The One Big Clock for Everything. Terry Pratchett has a delightful parody of this in Thief of Time. ]
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
hi Mike,
I have looked only briefly at all of your last replies. Right now I am reading Box 1.6 "Curvature of what?" of MTW. Then I will read and reply to your latest set of posts in this thread. BTW, thanks for all the help. Conceptual discussions of physics are a very good and rare thing - so they should always be appreciated. But w/o formalism, they can lead ... any where - good or bad - true or false - and so be leading or misleading. Physics is both equations and concepts. It requires both to have a good (concepts) and solid (formalism) understanding of physics.
When I was an undergraduate student I was told that MTW is called the "Princeton Phonebook" b/c of all the Princeton professors it references. I referred to it as "MTW's blerb about GR" as a joke. Joke may be on me b/c MTW is half as long as the Bible ... and it took me ~3 years to read that.
Anyway, page 32 of MTW says that Riemann "... spent his dying days at 40 working to find a unified account of electricity and gravitation." This didn't gel with your description so I went to Wikipedia.org for his bio to clarify. This is from their page on Georg Friedrich Bernhard Riemann
Austro-Prussian War
Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.[1] He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.
British TV: I've been a fan.
Wallace & Gromit, Jeeves and Wooster, Carry on Regardless, MPFC, The Rise and Fall of Reginald Perrin, Fawlty Towers, Blackadder, The Young Ones, RED DWARF (the first 6 seasons only), almost any Peter Sellers movie, Shaun of the Dead, Are You Being Served, Yes Minister, Yes Prime Minister, Rumpole of the Bailey, ... the list goes on.
-joe
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-LD
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my faith
I have looked only briefly at all of your last replies. Right now I am reading Box 1.6 "Curvature of what?" of MTW. Then I will read and reply to your latest set of posts in this thread. BTW, thanks for all the help. Conceptual discussions of physics are a very good and rare thing - so they should always be appreciated. But w/o formalism, they can lead ... any where - good or bad - true or false - and so be leading or misleading. Physics is both equations and concepts. It requires both to have a good (concepts) and solid (formalism) understanding of physics.
Yah. So true. I tend to play alot with the maths, more like shooting baskets in the back yard rather than properly playing a whole game on a court. I need more formal rigour .... but I also dislike what I can't put ( eventually ! ) into everyday words. A difficult tension. :-)
When I was an undergraduate student I was told that MTW is called the "Princeton Phonebook" b/c of all the Princeton professors it references. I referred to it as "MTW's blerb about GR" as a joke. Joke may be on me b/c MTW is half as long as the Bible ... and it took me ~3 years to read that.
Well, it's way thicker than our local phone book ( City of Melbourne )!! :-)
Anyway, page 32 of MTW says that Riemann "... spent his dying days at 40 working to find a unified account of electricity and gravitation." This didn't gel with your description so I went to Wikipedia.org for his bio to clarify. This is from their page on Georg Friedrich Bernhard Riemann
[quote]Austro-Prussian War
Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.[1] He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.
Err, I was talking about the bio of Schwarzchild wasn't I ??? :-)
But yes, he was a brilliant one lost way too young also .... :-(
British TV: I've been a fan.
Wallace & Gromit, Jeeves and Wooster, Carry on Regardless, MPFC, The Rise and Fall of Reginald Perrin, Fawlty Towers, Blackadder, The Young Ones, RED DWARF (the first 6 seasons only), almost any Peter Sellers movie, Shaun of the Dead, Are You Being Served, Yes Minister, Yes Prime Minister, Rumpole of the Bailey, ... the list goes on.-joe
Wow, I match 4/5 of those..................... with apologies to other countries, the Brits are by far the funniest in my view. Little Britain is the latest gem ... 'computer says no' ..... :-)
Cheers, Mike
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
In laymans terms. How is geometry linked to measuring force?
How is Time included in measuring the volume of a cube?
I always thought measuring velocities of cannonballs was physics.
How does one add 'time' to a geometrical coordinate?
Secondly : when we say geometry it's crucial, though no doubt annoying, to have to consider the basis of measurement. As Einstein found out with Special Relativity, some 'obvious' ideas in classical physics turned out to be wrong, or at least misleading and approximate.
So finally back to Newton's case of a single central mass that influences another one some distance away - your originally query. The good & bad news is that we have an approximate but not exact solution to the GR equations for this. It's still pretty good and has been observationally well confirmed more than a few times though ( eclipses, Mercury .. ).
So, Einstein's "obvious" ideas in classical physics were misleading and approximate, and there is observational evidence to support your theory, yet your theory is still approximate? How does that differ from the original problem Einstein had?
Edit: Saw a blackhole mentioned. Hasn't Crothers exposed bad maths concerning those? Since they are only a mathematical prediction.
http://www.nowpublic.com/tech-biz/mathematician-claims-black-holes-due-faulty-math-follow
You seem to be adequate with mathematics, so by reading that article you should come to the same conclusion.
In laymans terms. How is geometry linked to measuring force?
That's indeed the trick, there's no 'obvious everyday' answer. :-(
How is Time included in measuring the volume of a cube?
Not especially. *
I always thought measuring velocities of cannonballs was physics.
Certainly is.
How does one add 'time' to a geometrical coordinate?
Time becomes a geometric co-ordinate.
So, Einstein's "obvious" ideas in classical physics were misleading and approximate
No, Einstein corrected ( even ruled out ) some incorrect classical ideas produced by others. By 'incorrect' I mean those that contradicted experiment.
and there is observational evidence to support your theory, yet your theory is still approximate? How does that differ from the original problem Einstein had?
I don't have a theory. I'm trying to understand Einstein's, and in doing so I'm sharing my thought's on that.
Edit: Saw a blackhole mentioned. Hasn't Crothers exposed bad maths concerning those? Since they are only a mathematical prediction.
http://www.nowpublic.com/tech-biz/mathematician-claims-black-holes-due-faulty-math-follow
You seem to be adequate with mathematics, so by reading that article you should come to the same conclusion.
Sorry, I'm focused on Einstein's GR at present. In any case one would rule black holes in or out based on observation. If the maths is faulty then that needs adjusting. There's an oft missed logical truth - a false premise can imply anything.
Beg my forgiveness, but I tend to chuckle when a phrase like 'only a mathematical prediction' is used. I could use maths to predict what happens when you fall off a cliff. The outcome goes well beyond mathematics though .... I think what you mean is that we haven't yet 'seen' a black hole. Careful thought ought reveal an immediate difficulty there, but having said that there is no shortage of observational evidence upon the core of our galaxy ( probably our closest and best candidate ) for which we have no better conclusion of to date. But I appreciate your point. Science never gets it perfectly right, and never will. What is strived for is to obtain progressively better understanding of the world based on careful experiment, in a manner that selects the best from alternate explanations.
Aside : In fact while looking at MTW ( Box 1.10 on page 41 ) I note a quote of Newton's :
For hypotheses ought .... to explain the properties of things and not attempt to predetermine them except in so far as they can be an aid to experiments.
I reckon this is still exceptionally relevant today, when quite a slab of what is held under the umbrella of 'science' utterly ignores this, to wit : experiments/observations are all too frequently designed to confirm some prior preference/expectation and not enable a choice between competing explanations including diametric opposites if possible.
Generally, one great theme of GR is that it is possible to have a workable theory that predicts observational findings by viewing certain aspects of nature with geometrical ideas. Including bits that we previously wouldn't have thought of doing. Einstein was not the first to consider this, but he was the first to get it pretty right - to date as yet contradicted by observation or bettered by an alternative.
Cheers, Mike.
( edit ) * 'Not especially.' Actually you could have a cube in spacetime, although one might have to accept a cube with four dimensions - width, length, height and persistence. :-)
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
( edit ) * 'Not especially.' Actually you could have a cube in spacetime, although one might have to accept a cube with four dimensions - width, length, height and persistence. :-)
Philosophy has no place in physics ;) Either do abstract ideas and vague concepts.
Observing black holes, i get the drift. Observing predicted effects of...better statement.
I reckon this is still exceptionally relevant today, when quite a slab of what is held under the umbrella of 'science' utterly ignores this, to wit : experiments/observations are all too frequently designed to confirm some prior preference/expectation and not enable a choice between competing explanations including diametric opposites if possible.
Good to see you have an open mind to science. Many people seem to have forgotten what you stated above. Science has IMHO become like a religion. Too many things are not allowed to be questioned, even if obviously wrong.
Thanks for the reply.
Philosophy has no place in physics ;) Either do abstract ideas and vague concepts.
The full title of Newton's masterpiece the 'Principia' includes the term 'natural philosophy'. I think he was essentially wanting to push his contemporaries to actually look at the world they are in, rather than purely examining their own 'inner' world. Tycho Brahe had a head start on this with regard to questions and theories upon the state of the cosmos. His breakthrough was not any terrific model, but the brilliantly simple idea of going out and measuring to an as extreme degree as possible - and then see which ideas can withstand that. He devoted his adult life to that program by constructing and operating a decent observatory - Uraniborg in Sweden. Kepler then came along and built his neat theory upon that large body of data, which is still true to the degree of approximation it represents. Had Tycho not been as obsessively particular as he was, then ellipses and circles would have remained indistinguishable.
Good to see you have an open mind to science. Many people seem to have forgotten what you stated above. Science has IMHO become like a religion. Too many things are not allowed to be questioned, even if obviously wrong.
Thanks! I see the main ( social ) mechanism is the misuse of the label 'science' like a brand name. You keep the outer wrapping but the box is actually empty.
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Philosophy has no place in physics ;) Either do abstract ideas and vague concepts.
Without these, physics or (for that matter anything) could not move forward..
Without these, physics or (for that matter anything) could not move forward..
Yes, it could. The old fashioned way. Observe first, then explain it. If you can't see what something is doing, you can't really know, can you?
Multiple universes, i call that a vague concept. How do you explain going from 1 universe to multiple universes? Vaguely, and incoherently, with a bit of mathematical religion thrown in.
M-Theory, i call that an abstract idea. And lots of incoherent mathematics proving a theoretical view of what's not really happening.
Are either helping us with physics? They help me move forward with entertainment, i'll give them that.
I would only recall what Albert Einstein said:
"Theory determines what can be observed".
On the multiple universes idea I agree it is nonsense and I am taking part in a project called QuantumFire by the Cambridge Cavendish Laboratory which strives to undermine the Copenhagen interpretation of quantum mechanics in favor of the pilot wave theory of De Broglie and Bohm just to eliminate such nonsense.
Tullio
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What we observe and perceive through our senses is the baseline. All the rest is just relationships. If what we observe or perceive is wrong or incomplete then our relationships could be wrong or incomplete. Therefore I learnt never to hold on to anything to tightly. Including.. 1+1 =2. :-)
Including.. 1+1 =2. :-)
At least that one can be logically proven :P (of course it depends entirely on our definition of the symbols; both the syntax and semantics)
Yes, it could. The old fashioned way. Observe first, then explain it. If you can't see what something is doing, you can't really know, can you?
Multiple universes, i call that a vague concept. How do you explain going from 1 universe to multiple universes? Vaguely, and incoherently, with a bit of mathematical religion thrown in.
M-Theory, i call that an abstract idea. And lots of incoherent mathematics proving a theoretical view of what's not really happening.
Are either helping us with physics? They help me move forward with entertainment, i'll give them that.
Theory and experiment ought be cyclic and iterative. The trouble with theory is that it can continue without ever drinking at the well of reality. The trouble with experiment is that it can remain forever literal and anecdotal without generality. However you've chosen a couple of good candidates for shooting at the earliest opportunity ( in my view ).
Testing alternate universes would require ducking nextdoor to an adjacent one, and popping back to compare results, so we'll need to await either the occult or the right technology for that. See Terry Pratchett and his 'trousers of time' concept, in Night Watch say.
As for M-theory and string theories overall, I became rather discouraged after reading Lee Smolin's The Trouble With Physics where he firmly but delicately outlines the dreadful error of large swathes of physics faculties around the globe. Not one shred of data to guide them. No physical predictions emitted whatsoever. An impending risk of becoming even mathematical rubbish, as they'd not attempted to prove a conjecture in the late 80's crucial to the subsequent development of the field. The flower of an entire generation of young physicists quite wasted.
Now as for 1 + 1 = 2 here we are on solid ground. I'll relate a particular approach that I treasure. Let
0 = 'zero' = the null/empty set = {}
then
1 = 'one' = the set that contains '0' = {{}}
and
2 = 'two' = the set that contains '1' = {{{}}}
then
3 = 'three' = the set that contains '2' = {{{{}}}}
.... ad infinitum/nauseum. This easily yields the concept of a 'successor' - a set that immediately contains another. From this one can proceed as follows :
x + 1 = the set that contains 'x' = {x}
so the symbol sequence '+ 1' becomes 'take the successor of'. One still hasn't defined what 'one-ness' or 'five-ness' or 'five_hundred_and_thirty_seven-ness' etc mean, but you've converted cardinality ( associating numbers with sets ) to ordinality ( lining them up in a row ). It's simple, consistent and is built upon to produce the arithmetic we know and love. :-) :-)
Cheers, Mike.
( edit ) Dedekind had a brilliant concept of division of sets ( 'Dedekind cuts' ) to sort out the behaviour of the real numbers, especially giving exact meaning to the notions of 'supremum' and 'infimum' that are required for limit processes, and thus calculus.
( edit ) So you'd get 2 + 3 = 5 by first breaking the elements down :
2 + 3 = ( 1 + 1 ) + ( 2 + 1 ) = ( 1 + 1 ) + ( ( 1 + 1 ) + 1 )
= 1 + 1 + 1 + 1 + 1
here brackets are purely used as a way of grouping the symbols, even though they have other meanings added in later development. Then build back up to the answer :
1 + 1 + 1 + 1 + 1 = ( 1 + 1 ) + 1 + 1 + 1 = 2 + 1 + 1 + 1
= ( 2 + 1 ) + 1 + 1 = 3 + 1 + 1 =
= ( 3 + 1 ) + 1 = 4 + 1 = 5
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
At least that one can be logically proven :P (of course it depends entirely on our definition of the symbols; both the syntax and semantics)
If the proof survives everywhere in the universe and outside our minds and tools, I would hold on a little tighter:-)
If the proof survives everywhere in the universe and outside our minds and tools, I would hold on a little tighter:-)
That's exactly what Roger Penrose reckons in The Road To Reality. There's a Platonic existence of mathematics 'out there' waiting for us to discover! :-)
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Nice person, prof.Penrose. He even answered a latter of mine after me having sent him a draft of a paper "The coherent brain", written in 1980, which contained ideas very similar to those of his book "The emperor's new mind". He found my text "highly interesting" and invited me to read his next book "Shadows of the mind", which is very platonic.
Tullio
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Notes on '1-forms' or 'differential forms'
Strict definition : A 1-form is a real valued function upon a set of vectors that is linear in it's arguments.
Put a vector in, get a real number out. Multiply the vector by a constant, and the real value result also is multiplied by that same constant.
It's one way to generalise the idea of a 'dot product'.
It's the best linear approximation to the change of an independent variable when it's dependent variable alters.
Take a simple one-dimensional vector analogy eg. the real number line. Suppose I have a function ( this is not the 1-form ) that goes like the square of the number. Thus
f(x) = x^2
Now I know the first derivative of this : df/dx = 2x. This essentially becomes the 1-form as
the actual functional change of [ f(x) = x^2 ] by moving along by delta_x = delta_f ~ df/dx * delta_x = <df/dx | delta_x> ( in 'bra' and 'ket' notation )
where delta_x and delta_f are non-infinitesimal. The '*' multiplication is the one-dimensional version of the dot product. It is approximate '~' as there may be higher degrees of rates of change to apply. The full gore ( including other issues ) leads to a Taylor's expansion, but we toss away the higher terms here. The derivative used in the above is evaluated at some particular x, so the approximation holds only 'nearby' .... and of course is a better one the closer we are.
In several dimensions you get the 'grad' replacing the role of df/dx, and a vector with components like delta_x :
grad_f = [df/dx, df/dy, df/dz] = [d/dx, d/dy, d/dz](f)
v = [delta_x, delta_y, delta_z]
( various equivalent notations ). The grad components are partial derivatives - rates of change of the function f solely along the co-ordinate basis directions each in turn - evaluated at the point (x, y, z) of interest. The grad is not a number but something ( function/operator ) to be applied to what ? Yes, you guessed it, a vector. In this case v that represents a finite displacement.
the actual functional change of [ f = f(x, y, z) ] by moving along v = delta_f ~ df/dx * delta_x + df/dy * delta_y + df/dz * delta_z = [d/dx, d/dy, d/dz](f) . [delta_x, delta_y, delta_z] = < grad_f | v >
Couple of points:
- beware the 'sense' of the quantities ie. which way do things increase or decrease?
- we are not limiting the number of dimensions of the vector here. By extension we could have an infinite number of degrees of freedom.
- I say generalisation of dot product as one may not always be adding within expressions to get your real number result. So the Lorentz type metric ( working on relativistic 4-vectors ) has an opposite sign for the time co-ordinate slot compared to the spatial slots. { this has the curious property that it may evaluate to zero when the directions don't 'look perpendicular' - say when the worldline is that of a light ray, hence the phrase 'null geodesic' }
u = (u0, u1, u2, u3)
v = (v0, v1, v2, v3)
u.v = (-1) * u0 * v0 + (+1) * u1 * v1 + (+1) * u2 * v2 + (+1) * u3 * v3
- the 1-form doesn't give a field's functional value, only how it changes with a displacement. So with one electric charge in the field of another, a 1-form applied to a displacement would give the change in the potential ( energy ) say, not the absolute value with respect to some global reference. But you could still say "I start with some field value f(P) at a point P, make a little jump along a vector v and get a new field value which is approximately f(P) + <grad_f | v>"
Example 1 : the height of grass on an otherwise bare hillside. I walk around with 2 independent degrees of freedom, with a rate of change of grass-height from one position to the next ( this of course depends also on where I am ). I use a 1-form to calculate the how the grass height changes with each step.
Example 2 : calculate a quantum mechanical outcome by applying a 'bra' function to a 'ket' vector. I get a real number representing a probability within the range zero to one.
Finally, other nomenclature and phrases :
- a 1-form is a tensor of rank 1
- the vector is contravariant ( don't ask )
- the 1-form is covariant ( don't ask )
- the vector and the 1-form are dual ( to one another )
- applying the 1-form to the vector is a tensor contraction
Cheers, Mike.
( edit ) To save you worrying about asking .... :-)
Covariant / contravariant embodies the idea of 'paradoxical motion', an ancient idea. So if I'm driving North along the highway and I look at a cow in a paddock, the cow will seem to travel southwards compared to the distant background behind it. Conversely from the cow's point of view I'll be traveling northwards compared to the background behind me.
Similiarly if I change ( say by rotating or translating ) the co-ordinate system I use in a given circumstance then I could equally say the system was static but the ( electric/magnetic/gravitational ) field I was describing rotated/translated the other way. It is essential to realise that this must be true, because we ( believe we ) are describing an entity ( the field ) that has a reality apart from our arbitrary choice of a particular layout of rulers and clocks. If not, for instance, then I could change the ( external ) field to a new value simply by rotating myself around by 360 degrees!! We don't seem to live in a universe where this happens ..... :-)
( edit ) Sharp punters will note that this co/contra-variance business implies that if I have two nearby points P and Q then a tensor contraction ( a 1-form applied to the displacement vector between the two ) is an invariant across ( sensibly related ) co-ordinate systems. So if I take a step on my grassy hillside the outcome is independent of who is looking. This is comparable to the ordinary dot product of a vector with itself giving ( the square of ) it's length, again irrespective of a particular co-ordinate arrangement.
( edit ) 'the affine connection' : I see what you were getting at LD. Yes. We like affine transformations so that points in a line with certain relative separations remain in line with the same relative displacements after the transform. Sensible transformations do this, including Minkowski/Lorentz stuff. A shear ( different scale factors applied to orthogonal directions ) is affine. Say the case of the length in the direction of motion Lorentz contracting but not in the perpendicular ones.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
If the proof survives everywhere in the universe and outside our minds and tools, I would hold on a little tighter:-)
That's exactly what Roger Penrose reckons in The Road To Reality. There's a Platonic existence of mathematics 'out there' waiting for us to discover! :-)
or
Mathematics was constructed by the brain in order to be effective in this universe. :-)
Mathematics is music for the soul
Music is mathematics for the soul
(English saying)
Therefore they commute and form an Abelian algebra.
Tullio
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Mathematics is music for the soul
Music is mathematics for the soul
Ahhhh, that'd be J.S.Bach then ! :-))
Mathematics was constructed by the brain in order to be effective in this universe. :-)
Yup, the reason why maths seems to be 'out there' is because that's the part of the information blizzard we have evolved to match to. Those that didn't do so well at that contributed to the genetic persistence of predators instead. I don't think it is a fluke that initial explanations in physics ( historically ) relate to everyday kinematical/dynamical stuff - position, time, velocity, rates of change .....
Imagine if you could talk to a sentient bacteria ? They'd do loops around us discussing chemistry and talk your leg off on the topics of Brownian motion, diffusion gradients and crystallography. If you explained quantum mechanics to them, they'd say ( with a knowing smile and a chuckle ) "Oh that. Now, errr .... what was you're problem there? ". :-) :-)
The real joke of "Why did the chicken cross the road?" isn't the specific retorts. It's the assumption that reasoning/intent applies at all. Suppose a chicken has neither a 'road' concept nor that of 'sides'. A road shaped pattern in the visual field doesn't trigger neurology that has no need for it *.
Cheers, Mike.
( edit ) * - until recently that is. :-) :-)
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Another approach is the Peano axioms. You accept them and move on, with no deeper formal derivation being offered. But you wind up with what you want from arithmetic down the track. :-)
Equality is meant in the sense of symbols aliasing for the same thing with reflexive ( m = m ), symmetric ( m = n <=> n = m ) and transitive ( m = n and n = p => m = p ) properties.
Ordering then becomes an implicit consequence, so that one can proceed to unambiguously assign meaning to 'greater than' and 'less than' statements and symbology. Beware that 'greater than or equal to' ( ...etc ) is an umbrella term for two distinct cases which can't simultaneously hold - you just happen to have thrown a blanket/group label over them. The other annoyance is with inequalities and multiplying both sides by a constant : you have to reverse the inequality ( swap '>' and '<' ) if the constant is negative, and if the constant is zero as applied to a strict inequality then it doesn't follow ( zero isn't either less than or greater than zero ). This is especially relevant with purely symbolic variables like 'x', I have to consider the possible varieties of 'x' to avoid error. Anyhows 1 gets labelled as 'the first' and the Principle of Induction is locked in as #5 .....
The 'problem' with arithmetic is really zero coupled with the operation of multiplication. ANY number times zero is zero, thus the inverse ( division, or 'how many zeroes do you add together to get the number five' ) has no definition. Or put another way - if I tell you that the product of a non-zero number and zero equals zero, then you can't work back to uniquely tell me what the non-zero number must have been. It's not 'infinity', it's not more ( or less ) than any particular number you can state, it's just not a solvable problem. However, weaseling around this issue has lead to some superb stuff .... the limit process superimposed upon algebra being one path.
Cheers, Mike.
( edit ) And the other 'problems' - which number multiplied by itself gives negative one, and which ratio of integers gives the length of the longest sides of certain triangles - also have impressive solutions. :-) :-)
____________
"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
The real joke of "Why did the chicken cross the road?" isn't the specific retorts. It's the assumption that reasoning/intent applies at all. Suppose a chicken has neither a 'road' concept nor that of 'sides'. A road shaped pattern in the visual field doesn't trigger neurology that has no need for it *.
Cheers, Mike.
( edit ) * - until recently that is. :-) :-)
Interesting.. Is it possible to understand and view the universe rationally without "concept". I think the chicken does it quite well. and they get along amazing well together:-)
I remember playing the English Suites by JSB. Then I played an Olivetti Lettera 22 typewriter when I translated 12 books in order to pay for my mortgage and support my children. Now I only play a keyboard. Sob...
Tullio
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Now here's an interesting relevant bit o' jommetry I hadn't fully realised the import of before. The vector cross product makes sense in 3 dimensions only. For four or more, one can't uniquely assign a result by generalising the 3-D version. The cross product construct is involved in the Lorentz force law with electromagnetism say.
In 3-D the right ( or left ) hand rule lets you have a single direction which is perpendicular to the other two vector arguments. In 4-D there is an infinitude of directions which are orthogonal to any two given ( non-parallel ) vectors. So ..... if there are more independent spatial directions than are evident to us in this Universe we'd need some extra rules applying to explain why cross-product-dependent behaviours occur as seen by us. Or : what general rule applies in higher dimensions that specialises to the cross product in 3-D?
You see GR doesn't tell us what happens with other forces. Indeed one can use the difference between the world lines of a test particle with an electric charge compared to one without, in order to deduce that a non-gravitational force is acting at all ( preventing 'free fall' ).
Any/all of this may be solved already. Just musing ..... :-)
Cheers, Mike.
( edit ) Well, there's a trivial sense in which the above must hold. Extra dimensions by definition give one 'more room to play with'. I'm sorta getting at why, when GR can apply to very many dimensions ( at least I understand that to be the case ), the non-gravity forces are as they are. One for the 'Theory Of Everything' I guess ....
( edit ) Or I'm a chicken who is totally missing the relevant road and it's sides. :-)
Interestingly 'concept' is a concept in and of itself. It's a way of 'placing brackets', like the curly ones used for set definition :
{1, 2, 3, 4, 5 ..... }
The objects inside have a separate 'existence' but the act of bracketing them, thus grouping, is a concept. But I'll stop there. I feel madness lies along that path - all men are liars, barbers shaving themselves etc. Thank you Bertrand Russell.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
From what I remember, in a Hilbert space you have an inner product and an outer product. The first in a 3 space is the scalar product and the second is the vector product, which you call the cross product. A Hilbert space is a Banach space with an inner product. A Banach space is a linear normed space. All this from memory, so pardon any error.
Tullio
Edit: the linear vector space must also be "complete" to be a Banach space?
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Or : what general rule applies in higher dimensions that specialises to the cross product in 3-D? ... Any/all of this may be solved already.
Yes, I think Tullio's right about the outer product – I found a very nice tutorial here.
Regarding the question about the about the antimatter, there's some interesting news from several of Fermilab's ongoing experiments, namely MINOS, MinniBooNe, and D0 - -
At D0 it looks like they've identified a CP violation (charge-parity) with neutral B mesons – there appears to be an asymmetry between the way the B and anti-B mesons decay that deviates from the Standard Model prediction by 3.2 standard deviations.
At MINOS they measured the 'square of the difference between mass eigenstates' between muon neutrinos and tau neutrinos and came up with 2.35 x 10^-3 eV^2. A particle and its antiparticle should both have the same mass, right? When they measured the same difference between muon and tau anti-neutrinos they got 3.35 x 10^-3 eV^2. The statistical confidence is '2 sigma' – for the particle physicists, 3 or 4 sigma is pretty close to certainty, and in this case would mean some serious head-scratching. The scientists had to fire 7 x 10^20 protons at a target to gather the data for the muon/tau neutrino measurement, so it will probably be some time in early 2012 before recording enough data for 3 sigma confidence (assuming the effect is real).
At MiniBooNe the researchers have already weighed-in at 3 sigma confidence with their measurements of a similar mass difference for lower-energy muon neutrinos as they oscillate into electron neutrinos (compared to the muon and electron anti-neutrinos).
Wouldn't surprise me if spacetime turns out to be quantized and left or right handed ...
How do neutrinos get locked inside atoms in the first place, or are they created during the decay?
At D0 it looks like they've identified a CP violation (charge-parity) with neutral B mesons – there appears to be an asymmetry between the way the B and anti-B mesons decay that deviates from the Standard Model prediction by 3.2 standard deviations.
That's just "significant" by the usual measures. With a number of assumptions regarding behaviour of large numbers then 3 sigma 'means' that it is more than ~ 99.7 % likely that the effect of interest is not due to sampling variation ( a 'lucky streak' ). Or, if you like, if we repeated the experiment say, 1000 times we could attribute the ( same ) results from 3 ( 1000 - 997 ) of them due to random luck in sampling. The other 997 ( of 1000 ) we could not blame random variance upon. I don't really know why traditionally 3 sigma is the 'magic' line to equate to 'significance' ( I mean it's a fine choice for sure, but I just don't know the history of that ). Not all state it that way though, they just quote the sigma value and let others make up their own mind as to what level they will accept. 1 sigma is ~ 68.2 % and 2 sigma is ~ 95.4 % in normally distributed ( Gaussian ) statistics. 4 sigma is out at 99.99 % and each extra sigma from there on adds about another two 9's.
The biggest killer of this analysis is a systematic bias in measurement ie. an effect ( in your measurement system ) skewing the results in some direction. This can occur with radar guns, say - so to account for this ( or more likely to defray challenges in court ) in my state one is only sent a violation notice if the measured speed exceeds 3% above the allowed.
At MINOS they measured the 'square of the difference between mass eigenstates' between muon neutrinos and tau neutrinos and came up with 2.35 x 10^-3 eV^2. A particle and its antiparticle should both have the same mass, right? When they measured the same difference between muon and tau anti-neutrinos they got 3.35 x 10^-3 eV^2. The statistical confidence is '2 sigma' – for the particle physicists, 3 or 4 sigma is pretty close to certainty, and in this case would mean some serious head-scratching. The scientists had to fire 7 x 10^20 protons at a target to gather the data for the muon/tau neutrino measurement, so it will probably be some time in early 2012 before recording enough data for 3 sigma confidence (assuming the effect is real).
The main current model is that a mass eigenstate is an un-observable quantum state that when mixed with others gives an actual detectable neutrino. The idea is that as these eigenstates propagate they wax and wane so that the detectable mix ( which is what determines observation as an electron, muon or tau neutrino ) varies as you go along - from the Sun ( source ) to the Earth ( target ) for instance. Even worse, 'cos of quantum, a given eigenstate mix at some particular position only gives probabilities for observing a given detectable type. To deduce these probabilities/fractions one has to collect sufficiently large numbers of data points. Thus some eigenstates will be more likely to give electron neutrinos say ( if they interact at all ! ) rather than either of the other two. Mathematically this can be conceptualised in a vector space containing eigenvectors which have angles between them that specify the exact mixing character. The first was named after Nicola Cabibbo who first saw this connection ( ie. where would physics be without Italians, eh Tullio ? ) but generically all three are usually called the Cabibbo Angles. Well it's three if you believe there are only three neutrino types, which is another story again .... :-)
Anyway the consequence, via this scheme, is that it is OK to have a difference b/w muon neutrino and tau neutrino HOWEVER that ought be the same difference as b/w muon anti-neutrino and tau anti-neutrino. Why would the eigenstates not be invariant here ?! So yeah, if this firms up to 3+ sigma level - the scientific analog/version of the criminal/legal proof standard 'beyond reasonable doubt' - then the feline will well and truly be among the avians!! :-)
At MiniBooNe the researchers have already weighed-in at 3 sigma confidence with their measurements of a similar mass difference for lower-energy muon neutrinos as they oscillate into electron neutrinos (compared to the muon and electron anti-neutrinos).
So ditto but more so as their methodology is rather different. Looking 'real' then .... :-)
Wouldn't surprise me if spacetime turns out to be quantized and left or right handed ...
It'll be subtleties like the above that will define that, if at all.
How do neutrinos get locked inside atoms in the first place, or are they created during the decay?
Well, I think/remember a neutron as being "equal" to a proton plus an electron plus a neutrino. But that's a 'black box' answer, a rule of thumb, as we only know what we measure. Who knows what they get up to when we are not looking ? :-)
Cheers, Mike.
( edit ) So the percentages aren't a statement about % 'right' or 'wrong' but about the likelihood of a result being due to a random sampling, being taken from a large possible population ( here infinitely large ). Of course if we could measure/know every interaction in the universe for all time then we'd have no need for statistics : you'd just state what happened with that complete knowledge. Failing that we make some reasonable assumptions - basically that the universe is not perversely structured simply in order to fool us. :-)
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Thanks Mike, also in the name of Nicola Cabibbo who was unjustly denied a Nobel Prize last year. I have personally known two Italian physicists who deserved a Nobel Prize and never got it, experimentalist Giuseppe "Beppo" Occhalini (who was one of the very few Italian professors who refused to take a loyalty oath to Mussolini and exiled himself in Brazil where he earned his money by acting as a mountain guide) and theorist Tullio Regge, of Regge pole fame. But to earn a Nobel prize you must speak English.
Tullio
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One other thought on the matter/anti-matter issue : might the quoted asymmetry be equivalently stated as the weak force having a different distance dependence according to matter/antimatter - because it is the vector bosons ( Z0, Z+ and Z- ) mediating this behaviour. This could be 'seen' in the micro as differing masses and decay modes? But in the 'macro' perhaps this gives rise to an apparent 'force' that is evident as 'cosmic expansion'? Hoyle and others had a similiar argument in the late 1950's referring to an ever so slight difference in the magnitude of the proton vs. electron charge. Here about a 10^(-18) difference would be sufficient to explain the then measured expansion rate ( Hubble factor ). The idea fell over though, as direct measurements discounted the charge asymmetry to much lower levels.
The point is that dark energy/matter arguments pre-suppose that our knowledge of forces is complete to all scales. That is : we assume our best about these forces and when observation doesn't fully agree we are tempted to 'invent' new things. But it might be an inexact understanding or improper extension. The conditions in the early universe our well out of current experimental range, we can only examine relic data and are thus limited by the frailties of that approach. Maybe it's not only gravity that needs a fresh look at with large scale ...
Cheers, Mike.
( edit ) I think it is the cosmologist Edward 'Rocky' Kolb who refers to dark matter/energy as 'epicycles', thus alluding to the tacking on of more circles to a theory already full of them. What was needed, per Kepler, was an ellipse. He's a wag for sure, so have a glance at this hilarious but shameless machine-gunning of political correctness from the late 1990's. :-) [ By that I mean : in Australia he would have been successfully prosecuted for racial/religious/ethnic/minority 'vilification' .... ]
( edit ) Addendum. How could anti-matter be involved in cosmic expansion when all we 'see' is matter? The key would be the extraordinarily low cross-section ( chance ) of interaction with neutrinos, of any type. You could have a universe brimming with anti-neutrinos and hardly feel it directly.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Here's an aspect of Minkowski geometry that deserves some mention. In Special or General Relativity you can actually have a metric of type :
provided that you accept an imaginary time co-ordinate. That is instead of :
time = t
you have :
time = i t
where t is a real number in both instances and i is the square root of minus one. Yes, complex numbers. This approach seems to be either praised or deprecated depending upon who uses it eg. Hawking is quite keen on imaginary time, but not MTW say.
A significant problem with the ( historically based ) words 'imaginary' and 'complex' is that they impart a sense of dread, or expectation of difficulty, when first encountered. As Penrose in 'The Road To Reality' repeatedly outlines, the real world at small scales is superbly and efficiently modelled by using these mathematical entities. With GR we are taking the limit of 'nearby' points, that is all spacetime geometry obeys Lorentz in the small. And they really aren't as hard to understand or use as they are often made out to be.
So the spacetime 'distance' differential still winds up to be :
ds^2 = - dt^s + dx^2 + dy^2 + dz^2
because i^2 = -1. Now apart from being a spot of clever algebra, where has this got us? We've converted :
(A) A 'space' with four real number components ( t, x, y, z ) and a Minkowksi metric.
to
(B) A 'space' with one imaginary and three real number components ( it, x, y, z ) with a Euclidean/Pythagorean metric.
Both have the same differential .... so the classification of world lines into time-like ( normal sub light speed ), space-like ( never seen or dis-allowed tachyonic ) and null ( light-like, at the border between the first two ) is unchanged. So subsequent theory - mechanics, interactions etc - don't change. Are we so desperate to stay with a classical metric that we go for a strange time definition?
This has annoyed me for several years. Here's my rough take on things, for what it's worth. I'm shooting the breeze on the night shift 'cos the traffic is slow ( and wet ). See, I've even got time to stuff about with BBCode, especial thanks to the new E@H web interface !! :-)
I've listened to Hawking in his resolution of singularities by the use of imaginary time. Generally speaking one can use complex extensions of real number functions to avoid poles of functions - where some are expressed as the inverse of some polynomial, and said polynomial has zeroes at certain points so that the inverse is 'infinite' or more properly simply 'undefined'. The poles are still there in the Argand plane but one can 'drive around them' eg. with contour integration. Integration is a key mathematical tool as it adds up lots of little changes ( like the progression of a particle's proper time ) over some history, world-line or whatever. The twins in the Twin's Paradox get 'integrated' over different spacetime paths for instance, to find out how their ages differ when they meet again.
So if you want to integrate a function from a negative real ordinate to a positive real ordinate, but only along the real axis when there is a pole at zero ( say 1/x ), then you can't legitimately do that. But if you extend 1/x to being complex ( 1/z ) in the right way ( 'analytic' ) you can do the integration ( with some reasonable cautions ) and avoid the pole at the origin. If done correctly the integral will have a value ( total ) that is independent of the precise path along which integration is performed. Also Penrose gives a beautiful example of how the real roots of some cubic equations can be derived 'easily' ( or more so than otherwise ) by allowing complex numbers as intermediaries ( conjugates are taken to come back to the real line ).
So where are we going here? More clever algebra? Think of what it means to have an 'imaginary' ordinate. Take the Argand plane, reals along the x-axis and imaginaries up the y-axis. Grab any point on the plane at all, except the origin, and multiply it by i. It will be rotated around the origin by 90 degrees ( anti-clockwise, but that is simply by convention ). Do that twice and you reflect through the origin ie. z becomes -z. Do this reflection twice and you are back where you started, where indeed all these rotations are a group with a 2PI modulus. Sound even vaguely familiar?
Yup, spin. Specifically fermions and bosons, and the fact that by exchanging indistinguishable particles you can either subtract from quantum sums ( leading to exclusion of Fermions from identical states, due to 'opposite' phase ) or add within those sums ( leading to clumping or condensation of bosons, due to 'same' phase ). You could transition between fermionic and bosonic behaviour purely by pushing the 'multiply the time ordinate by i' button twice. Spin is a time rate of change, or at least can be linked to measurable macroscopic quantities like angular momentum which demonstrate that ( magnetic resonance imaging is a good example of this ). Quantum mechanical phase is never directly measurable, only differences thereof. I can use either the (A) or the (B) scheme above to represent reality, leaving all results known to date unchanged ( as the spacetime differential is invariant to the choice ).
So where are all these 'super partners' that we'd love to have around, in order to stop our integrals hairing off to infinity, instead of something sensible, when we want to sum over virtual particle interactions? I reckon they are always there but we'll never get to meet them directly. They are only one push of the i button away. They don't appear in our 'measurable' metric, by having a real time co-ordinate in our Euclidean metric ( or an imaginary one in the Minkowski. Choose either view ). But maybe they ought appear in the 'intermediate' calculations, even if only to be 'nulled' by conjugation prior to the final result ( recall that the complex conjugate of i is -i ).
Maybe the super-partners aren't distinct particles at all, just in a different part of a 'super phase' cycle. Instead of phase having a real value like the classic one dimensional back and forth of a pendulum swing, a super-phase would actually be complex and follow a unit circle ( centered on the origin ) in the Argand plane. We only 'see' the super phase at it's projection onto the imaginary axis ( or maybe the real ? hmmmm ) .... and call that component 'phase' by the usual meaning.
Silly me. Churning bandwidth and musing. :-)
Cheers, Mike.
( edit ) I wouldn't think the above is especially original. No doubt someone has already tried, and probably failed, along this track ..... maybe it abrogates causality or somesuch.
( edit ) Aside : Say I stick with a Minkowski metric but convert t to it - just for a single particle ( cheeky ). ( This is not either A or B above ). Then I convert a time-like ( detectable, sub light speed ) world line into a space-like ( tachyonic, undectectable ) one. For us the particle 'disappears'. Now take a tachyonic particle an 'rotate it in' by going from it to t. It 'appears'! Why? Well, it was always 'there' in spacetime but not in our light cone ( past or future ). Maybe this is how you 'create' and 'destroy' these guys. You don't have to talk of converting mass to/from energy ( though it looks that way in a given light cone ) you are just flipping them in/out of view. Mass/energy is still conserved. Remember 'mass' for us is a way of saying 'it should stay in our light cone'. :-)
( edit ) To be more precise you can swap b/w normal and tachyonic particle 'type' by going from a real velocity to a pure imaginary one. If you accept that the spatial co-ordinates don't change character, then you must attribute that change as being due to going over to imaginary time.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
The main difference I know between bosons and fermions is that you can quantize a boson field with commutator [ab - ba] and a fermion field with anticommutators [ab + ba]. This was the main theme of my thesis in 1966. The part related to the boson field was published in Il Nuovo Cimento in 1967 (G. Bisiacchi et al; I was the al}. The second part was never published but I found some of its ideas in an Internet paper by the American physicist Mike Guidry called "Fermion Lie algebras and Microscopic Theories of Nuclear Structure". Of course he did not know my thesis but I sent him a copy of it with an English translation of my second part. I then lost my Internet connection and do not know his reaction. But if you do not publish (which is a costly matter) you lose all priority.
Tullio
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I found a great little book, originally published in 1945, that teaches the mathematical basis of general relativity and touches on the main point of this thread - defining gravity. I highly recommend this book which is written as a Socratic dialog with plenty of really cool drawings.
The Einstein Theory of Relativity: A Trip to the Fourth Dimenion, by Lillian R. Lieber.
http://astore.amazon.com/amuletdevelopmco/detail/1589880447
After several false-starts, and about 20 introductory books later, I can now say general relativity is mine thanks to this book. Tenors rule!
Daniel
The Physics Groupie
Congrats Tullio for being the al in et al. I'd always wondered who that was!!
Alas ( per Google etc ) it seems Il Nuovo Cimento was either discontinued or superceded in 1971. As you say, publish or perish.
G'day Dan! Thanks, I'll go out and order that one. Always looking for a fresh approach.
.... about 20 introductory books later ....
I know the feeling. I've had a lifelong ( well, since my teens ) fascination with maths. Especially calculus. So I have Pre Calculus, Introductory Calculus, Calculus of One Variable, Calculus of Several Variables, Calculus To Read On The Train, Intermediate Calculus, Deleted Neighbourhoods Are Really OK, Advanced Calculus, Limits : Why Bother ?, Postgraduate Calculus, Introductory Real Analysis, Frontier Points - How Far Would You Go? , Introductory Complex Analysis, How To Close A Difficult Open Set, The Low Down On The Supremum, Real Analysis on Weekdays, The Not So Mean Value Theorem, Complex Analysis For A Long Weekend, Schaum's Outline of Differential Equations, Making Room For Accumulation Points, Continuity For Remote Students, Solvable Integro-Differential Equations ( total of five pages, including the title, preface, contents and index ), My Life As a Differential : The View from the Slope, A Quick Recipe For Lime Soup, Grow Your Own Compact Metric Space At Home, Calculus for Physicists Or Why You Can Cross Multiply by dx, Integration - Yes, Life Is a B**ch, What Newton and Leibniz Really Thought Of Each Other and my all time favourite You Have To Get A Computer to Numerically Solve It Anyway .... :-) :-)
Cheers, Mike.
( edit ) So Penrose's The Road To Reality is for me the pinnacle. Mind you the any of Schaum's series ( McGraw Hill ) are superb for hacking through the forest. You have to practice you see, to really get it, and their guides have long ( but graded ) problem sections with worked solutions. If you're really serious about a future in maths then you couldn't go ( far ) wrong with a bunch of pens ( I prefer 'felt' tipped ~ 0.3mm and various colors ), a stack of writing/graph pads, and a Schaum's book to plow through!! :-)
( edit ) Do not worry about your difficulties in mathematics, I assure you that mine are greater - Albert Einstein
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Hi Mike,
And G'Day to you too.
I'm a big fan of Road to Reality as well. It is a nice book to see how the pieces fit together. Can't go wrong with Penrose.
If you're a fan of Calculus then you MUST have "div grad curl and all that" by h. m. Schey. This is a classic and has a lot of physics in it too. Just a small paperback, it is something you can stick in your bag for a long train ride. I read it on the plane to get that warm and fuzzy feeling that maths always brings me. Enjoy.
Daniel
The Physics Groupie
Hi Mike,
And G'Day to you too.
I'm a big fan of Road to Reality as well. It is a nice book to see how the pieces fit together. Can't go wrong with Penrose.
If you're a fan of Calculus then you MUST have "div grad curl and all that" by h. m. Schey. This is a classic and has a lot of physics in it too. Just a small paperback, it is something you can stick in your bag for a long train ride. I read it on the plane to get that warm and fuzzy feeling that maths always brings me. Enjoy.
Cool. I'll look that one up too. Good old Amazon! => Indeed one reviewer for that states :
It's been over two decades since I first studied vector calculus from my old textbook on electromagnetic fields and waves (Lorrain and Corson, Freeman, 1970).
That book, Electromagnetic Fields and Waves, is precisely where I first touched the subject.
Another nifty pocket size one for the road ( train/plane/bus/hovercraft/interstellar-shuttle/matter-transfer-cabinet ) is Metric Spaces by E.T. Copson. It's one of the less brutal from the Cambridge Tracts in Mathematics, which you generally either read by the fireside or throw in to keep yourself warm .... :-)
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
Mike, evidently Il Nuovo Cimento (nuovo because the old was the Proceedings of the Accademia del Cimento in Galileo's times) suffered from the onslaught of on-line publishing. My Nuovo Cimento article was in the B Series, dedicated to relativity and mathematical methods in physics. Unfortunately my coauthor Giordano Bisiacchi, who was my thesis adviser, died in a car accident in 1972 and that put a stop to my publishing. Asides from some chapters in Mondadori and Isedi Encyclopedias, which count very little in academia, my biggest contribution was the Physics and Astronomy part written in English for a report "Current trends in basic research" written under a contract with UNESCO in 1972. It was to be a part of a book by Italian biologist Adriano Buzzati-Traverso, brother of writer and painter Dino Buzzati, who unfortunately delayed the printing of the book. When printed in 1978, some parts of the report were obsolescent or obsolete, and the "Nature" staff, which had competed with us of Mondadori Scientific Publishing staff for the contract, were quick to point it out. Publish or perish, as you say. Cheers.
Tullio
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I've had a further look at the cross-product thingy, plus the imaginary time co-ordinate from The Road To Reality ( TRTR ). There's a deep topological theorem that places restrictions on the behaviour of quaternions etc .... giving the best ( most realistic ) results for 'spinors'. These model particles with spin BUT with an imaginary co-ordinate(s) that do for a 180 degree rotation what i does over 90 degrees. This implies that a 4PI rotation about an axis brings you back to the original state, so is sort of an identity transform.
This almost immediately reminded me of a neutron study done in the late 1970's by some chaps I knew at the University of Melbourne Physics Department ( on sabbatical to the neutron reactor facility in France ). They didn't unload anything ground breaking but fairly directly and cleverly showed the interference issue with neutrons ( as fermions ).
Recall that spin-1/2 particles have to be turned around twice, at 360 degrees per turn, to come back 'the same'. Generally if a particle is spin-X then 360/X is the required least number of degrees to turn ( X non-zero ). A photon is spin-1 thus if, say, you have one coming towards you with a certain polarity ( the electric/magnetic field orientation ) and then you circle around it once ( 360 degrees ) that polarity looks unchanged. However a ( hypothetical ? ) graviton is spin-2, and this gives rise to two independent directions of polarization 45 degrees apart. Spin-0 by the way is like a perfectly smooth featureless sphere, you can't decide upon any special orientation regardless of your direction of view. There's considerable subtlety with the spinor construction, but I see that it is essential to get it correct - quantum mechanical calculations have no hope of coming out right otherwise.
Beware of the two uses of the word 'spin'. The first ( that we are referring to above ) is the total spin vector, which we can't actually measure per se but models 'as if'. The second is the component of that vector that we measure along some direction - while then remaining clueless about the components in the other two perpendicular directions.
Penrose doesn't like imaginary time, because it gives illusion of a 'proper' Euclidean metric.
Now I think I've found Tullio's effort : "SU n,1 Representation for the Harmonic Oscillator", G.Bisiacchi and T.Chersi. Il Nuovo Cimento B Series 10 ( 1967 ), vol. 51, issue 1 pages 195-198 ???? Which unfortunately SpringerLink won't allow prying from their fingers for less than $40 USD. :-(
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
I still have a printed copy of it. If you PM me a post address I might photocopy it and send it to you. We had a good number of requests for it, one from the Academy of Sciences of Kazakhstan. Cheers.
Tullio
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Thanks! Actually if you PM me your address I'll send you a stamped self addressed post-pack type pre-paid priority etc thingy to return it in! I'd be delighted to read your efforts with T.Chersi at al :-) :-)
Another view on spin zero. The 360/X doesn't agree with the pattern of the others, as if you take X > 0 but tending to zero as an infinitesimal - that should give an unbounded ( tending to infinite ) value for the number of degrees to rotate and get the 'same thing'. Now if you take the converse case, the spin X tending to infinity hence 360/X goes to zero - all infinitesimal rotations are now giving the 'same' thing, which is more realistic for spin zero. So that suggests a duality b/w zero and infinity ( or unbounded and infinitesimal ), which is key real estate for the mapping from complex plane to complex plane via the Riemann sphere ( inside |z| = 1 goes to outside |z| = 1 and vice versa. ). The extended complex plane that is, with one way of the mapping you project from the 'south' pole at (0, 0, -1) : hitting the 'northern' hemisphere for |z| < 1, the 'equator' for |z| = 1 and the 'southern' hemisphere for |z| > 1. The real line becomes 'painted' to a great circle through both poles and the pure real numbers at (-1, 0, 0 ) and (+1, 0, 0 ). The inverse mapping is done mutatis mutandis with projection from the 'north' pole at (0, 0, +1). The other nifty bit is the implicit complex conjugation ie. i <-> -i, so one is not only going to the other 'side' ( inner <-> outer ) of the circle at |z| = 1 via the mapping but there is an orientation reversal of PI or 180 degrees. Can you see how these themes just keep knocking on the door ??? :-)
Cheers, Mike.
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"I have made this letter longer than usual, because I lack the time to make it short." - Blaise Pascal
An interesting take on Erik Verlinde Idea (Gravity is just a consequence of thermodynamics)
I have read a little bit. I like to read a little and digest. That way I don't get to fat :-)
Entropic Gravity Predestrians
An interesting take on Erik Verlinde Idea (Gravity is just a consequence of thermodynamics)
I have read a little bit. I like to read a little and digest. That way I don't get to fat :-)
Entropic Gravity Predestrians
I read about it in the NYTimes of July 12. By the way, look to Citizen Cyberscience Summit program in the BOINC home page. Looks like dr.Allen is going to give a talk on Einstein@home, together with other BOINC gurus.
Tullio
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Gravity theories abound. I have read on New Scientist of a theory by Petr Horava at Berkeley U. who thinks we should abandon Lorentz Symmetry. I don't think this is possible but since all these new theories cannot be tested by any experiment, I think we should go back to Newton who said "Hypotheses non fingo".
Tullio
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