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.

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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.

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

I'm relieved! It's a toughy to understand, much less explain! :-)

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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 ...

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! :-)

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.)

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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)

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- 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]?

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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!

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- 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."

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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]

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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]

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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?? :)

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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.

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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?

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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.

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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!

-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.

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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 ).

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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 :

[pre]-1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1[/pre]
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 ).

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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.

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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.

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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.

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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!! :-)

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Yes, on the light cone analogy. I think I am understanding it now.

Like a Hogwart's hat, it's roughly conical but squashy. :-)

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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 ?? ).

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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 ).

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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.

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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 !! :-)

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 ???? :-) :-)

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. ]

## Not Gravity, 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.

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.

## That's how I see it. Take a

)

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

## RE: It's the ( Karl )

)

Indeed it's Schwarzschild (black shield, not child ;-)

GruÃŸ,

Gundolf

Computer sind nicht alles im Leben. (Kleiner Scherz)

## RE: RE: It's the ( Karl )

)

Ah, so not 'the child of Schwartz' then. Thanks, I've seen many spellings. With the 't' is probably an anglicized mangling ... :-)

Cheers, Mike.

## RE: That's how I see

)

I'm relieved! It's a toughy to understand, much less explain! :-)

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.

## RE: RE: In GR, how does

)

Thanks for the PG Tips.

(http://www.youtube.com/watch?v=gfG2ZujlIZU&feature=related)

-LD

________________________________________

my faith

## I originally asked, "In GR,

)

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.)

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)

[tex]T_{\mu\nu}[/tex]?

OHhhh... I see. Since the mass moves in a way according to the spacetime!

Yes, b/c measurements have a certain precision. "Flat" really means "I cannot measure curvature below the precision of my equipment."

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]

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]

Right... we say they are in a "straight" line, but in reality they are following along the geodesic! Yes?? :)

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.

I thought it was exact. Was this the problem with the cosmological term?

Oh wow... I did not know that. What a shame.

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!

-LD

________________________________________

my faith

## RE: May I have some of the

)

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.

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 ).

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 :

[pre]-1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1[/pre]

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 ).

Yup, eta is the straight co-ordinate difference approach whereas gee is a function ( group thereof ..... ) that varies across your spacetime landscape.

Dunno. Probably :-)

All I know of that is the Killing vector tells you where to go to preserve/not-distort distances on an object.

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.

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!! :-)

Like a Hogwart's hat, it's roughly conical but squashy. :-)

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 ?? ).

Poor lad, got a crappy skin infection from the horrible mud of the trenches and died from blood poisoning ( no antibiotics then ).

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.

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 !! :-)

## A concrete but very

)

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 ???? :-) :-)

## RE: Question : could the

)

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.