Time & Gravity: GR in Physics Today?

[Dravish]

From my understanding of GR, time and gravity were shown to be essentially the same thing, is that the generally accepted interpretation in contemporary physics?

 

[robphy]

From my understanding of GR, time and gravity were shown to be essentially the same thing, is that the generally accepted interpretation in contemporary physics?

Contemporary physics does not interpret
Time and gravity as essentially the same thing.

What GR did is to assert that time and gravity are not generally independent concepts... they are related [in a specific way according to GR]... but they are not the same thing.

 

[Dravish]

In that case I must be missing something fundamental to the theory, how are gravity and time differentiated?

 

[anantchowdhary]

Gravity...is because of mass-energy and also other factors.Time is however not there because of mass

 

[Dravish]

To quote from ‘Time, Gravity and Quantum Mechanics, by W. G. Unruh’;
“A more accurate way of summarizing the lessons of General Relativity is that gravity does not cause time to run differently in different places (e.g., faster far from the Earth than near it). Gravity is the unequable flow of time from place to place. It is not
that there are two separate phenomena, namely gravity and time and that the one, gravity, affects the other. Rather the theory states that the phenomena we usually ascribe to gravity are actually caused by time’s flowing unequably from place to place.”

It seems that perhaps gravity and time should be equated (though I personally think that they are aspects of the same phenomena seen from 'different angles' so to speak), surely this will lead to an evaporation of many of the issues that arise when trying to unify QED and GR

Any thoughts on this line of thinking would be appreciated as I am trying to put together a paper using this perspective

 

[anantchowdhary]

I don't think time exists because of mass.But gravity,the curvature of space-time depends on the presence of mass.

So it can be said that

Only the flow of time is affected by gravity or by the presence of huge mass,energy etc.

 

[pervect]

What is behind what Unruh is saying is that one can often derive gravity from the metric coefficient g_00, a metric coefficient which in one common viewpoint is described as "gravitational time dilation". (I'd have to review the derivation to recall exactly which assumptions are needed to derive the "force" of gravity in this way, for starters one needs a static field, though. The force turns out the be proportional to the gradient, i.e. the rate of change with respect to position, of g_00 when the necessary assumptions are made).

However, it is possible and perhaps more accurate to take the view that time always flows at the rate of 1 second per second - in which case, it becomes more complex to describe exactly what g_00 is describing. We can say that it describes the "metric", of course, but that just invites other questions.

The usual way of saying this is to say that it is space-time curvature that causes gravity, not time that causes gravity. This approach can also cause arguments, however, which is perhaps why Unruh avoided it. (The issue here is that curvature is sometimes used losely to describe varying metric coefficients, but is othertimes used more strictly to indicate a non-zero Riemann curvature tensor. According to the first defintion of curvature, it makes sense to call gravity curved space-time, but it does NOT make sense if one uses the second, stricter defintion of curvature.)

Basically, the issue is one of semantics, but I don't think that it would correct to represent "time=gravity" as a standard view. (It might, from your quote, be reasonable to represent this as Unruh's view, I suppose - i.e. the view of one particular physicist, rather than a general consensus.)

To a certain extent, the problems of how to popularize GR are like the problems of how to describe an elephant to a blind man, i.e.

http://www.mcps.k12.md.us/curriculum/socialStd/grade7/india/Blind_elephant.html

so it isn't really too surprising that on the popular level, different descriptions of GR might appear to be different to the layperson when they are actually describing the same theory.

 

[Robert J. Grave]

Mass/energy and space/time together is the universe. They cannot exist seperately. The shape of space/time is related to the mass in it. All aspects of space/time are altered by this shape change. Three spatial and one time dimension. This effect is the gravitational effect. The more accute the bending or distoting of space/time the more pronounced the effects are. The black hole discriptions speaks of time stopping inside it.
This is the extream of space/time warping on to it's self. - Robert

 

[MeJennifer]

Unruh, in http://arxiv.org/PS_cache/gr-qc/pdf/9312/9312027v2.pdf" juxtaposes time in GR and time in Quantum mechanics.

In GR there is no background, particles do not live in spacetime they make spacetime, in other words, time and space are emergent properties of the gravitational field. Time is not like some infinite and perfectly flat bowling lane on which particles roll. Each observer can have a unique notion of time and space. This notion completely depends on his relative position in the combined mass and energy distribution of all causally connected particles and thus constantly changes.

In quantum mechanics it is quite different, here time is a background, here time is like an infinite and perfectly flat bowling lane on which particles roll.

 

[Robert J. Grave]

If time is flat in QM, how do we explane time dilation in relativity, an observed occurance? -Robert.

 

[pervect]

Mass/energy and space/time together is the universe. They cannot exist seperately. The shape of space/time is related to the mass in it. All aspects of space/time are altered by this shape change. Three spatial and one time dimension. This effect is the gravitational effect. The more accute the bending or distoting of space/time the more pronounced the effects are. The black hole discriptions speaks of time stopping inside it.
This is the extream of space/time warping on to it's self. - Robert

How would we go about testing these statements? It seems to me this is more a matter of personal philosophy than something that can be put to experimental test, though I'm willing to be convinced otherwise if you can demonstrate some way of testing your statements.

 

[jimbobjames]

Here are a couple of statements showing the link between time and gravity:

1) A clock at the top of a tall building ticks faster than a clock at the floor of the same building. This follows directly from the equivalence principle, by studying the behavior of clocks at the front and rear of an accelerated rocket.

2) The path which a freely falling particle takes is that for which the proper time of that particle is greatest.

Strap a watch to a tennis ball and throw it up into the air, remembering that clocks higher up tick faster than those lower down. The velocity of the ball causes the watch to tick more slowly (a special relativistic time dilation effect) but the higher the ball goes, the faster it ticks (since it is then higher in the gravitational field) - the net result of both effects (slowing due to speed and speeding up higher in the field) is that the actual path taken is the one which gives the most ticks on that watch. Any other path would result in less ticks. Its an optimization.

The beauty of the geodesic equation is that it includes the SR effect as well as the field effect into one succinct coordinate invariant equation.

"Straight line paths" through curved spacetime (geodesics) are those for which the proper time is greatest.

Of course it is still interesting to consider why it is that clocks at different points in the field tick at dfferent rates - does anyone out there actually understand why this is? Does a deeper study of GR or String Theory or QFT provide the answer? (Id prefer to know up front so I can optimize my time usage!).

(To say the answer is simply "spacetime curvature" would be circular, because different clock rates at different places is curvature of the time part of spacetime).

 

[pervect]

Of course it is still interesting to consider why it is that clocks at different points in the field tick at dfferent rates - does anyone out there actually understand why this is? Does a deeper study of GR or String Theory or QFT provide the answer? (Id prefer to know up front so I can optimize my time usage!).

(To say the answer is simply "spacetime curvature" would be circular, because different clock rates at different places is curvature of the time part of spacetime).

"Why" questions are always somewhat problematical, but here is an analogy which may help you.

Suppose you have a naval ship that can travel on the Earth's oceans at some constant rate - say 20 knots, where a knot is a nautical mile per hour.

At the equator of the Earth, 1 nautical mile is 1 minute of arc of longitude, so the speed of the ship is 20 nautical miles per hour, if the ship goes east-west.

Nearer the poles, at higher latitudes, however, the same ship, going at an identical rate of 20 knots, travels more than 20 minutes of arc per hour.

This is analogous to the way that time dilation acts. The ultimate cause is the same in both cases, in one case it is the curvature of the Earth, in the other case it is the curvature of space-time.

One way of describing things would be to say that the metric on the Earth's surface changes, and that distances are shorter at higher lattitudes.

But this may obscure the fact that a nautical mile is a nautical mile - it's really the coordinates (degrees of longitude) that change size, not the actual distances.

In GR, it's a similar situation. A second really is a second no matter where you go, but when you set up a coordinate system (similar to the degrees of longitude), you find that coordinate time intervals are not the same as the physical seconds one measures with a local watch.

 

[intel]

Unruh, in http://arxiv.org/PS_cache/gr-qc/pdf/9312/9312027v2.pdf" juxtaposes time in GR and time in Quantum mechanics.

In GR there is no background, particles do not live in spacetime they make spacetime, in other words, time and space are emergent properties of the gravitational field.

If this is so - particles make spacetime. Then what makes the gravitational field? I believed large masses were their cause, but small particles make big masses - so it still leaves what makes the gravitational field?

 

[jimbobjames]

Thats a good analogy, Pervect. Cheers.

The real physical time is the one on the watch, regardless of the coordinate system you choose to set up. Right.

"A second really is a second no matter where you go":

But the real rate of ticking really is different at different heights in the field, since if I bring those watches back to the same place later, the watch readings differ (regardless of coordinate choice). The relative number of physical ticks on the watches at different heights, which you can measure and compare when you reunite the watches, is different (and is independent of coordinate choice). And so one real second for me might correspond to 2 real seconds for you, it depends on our relative paths through spacetime. Or am I wrong about this?

I'd love to see a picture of the 2-d projection of the 4-d spacetime in the spherically symmetric (but weak) field near a planet. The 2 dimensions would be the radial (or height) and (coordinate-) time dimensions.

Geodesics on that curved 2-d surface - assuming the curvature to be the time curvature only - would show what we perceive as objects falling from large r to small r. What shape is that surface - can it be embedded into a flat 3-d space?

Have you - has anyone here - ever come across a visualization like that? I'm sure it would help me understand the analogy the relationship between gravity and time even better.

 

[pmb_phy]

From my understanding of GR, time and gravity were shown to be essentially the same thing, is that the generally accepted interpretation in contemporary physics?

I think it was Einstein who said it best when he said in Nature (Feb. 17, 1921, p. 783)

..., in respect to its role in the equations of physics, though not with regard to its physical significance, time is equivalent to the space co-ordinates (apart from the relations of reality).

Pete

 

[pervect]

Thats a good analogy, Pervect. Cheers.

The real physical time is the one on the watch, regardless of the coordinate system you choose to set up. Right.

"A second really is a second no matter where you go":

But the real rate of ticking really is different at different heights in the field, since if I bring those watches back to the same place later, the watch readings differ (regardless of coordinate choice). The relative number of physical ticks on the watches at different heights, which you can measure and compare when you reunite the watches, is different (and is independent of coordinate choice). And so one real second for me might correspond to 2 real seconds for you, it depends on our relative paths through spacetime. Or am I wrong about this?

No, you are not wrong. But you can set up an analogous situation in the analogy.

Naval ship Alpha starts at 0 degrees longitude on the equator, and goes east for 20 nautical miles. It winds up at 1 minute of longitude because, at the equator, one NM = 1 minute of longitude. (I may have gotten my signs wrong, maybe it's -1 minute of longitude, but it doesn't really matter.)

Naval ship Beta starts out at the equator, and travels north to 45 degrees latitude. It then proceeds east for 20 nautical miles. It then goes south back to the equator. Ship Beta finds that it has traveled MORE than 1 minute of latitude.

So, if degrees of longitude were time, if we image ship Alpha continuing on its trip along the equator, ship Beta would see ship Alpha as being "older" when the two ships unite.

You could explain it as "time" (distance) being differentat different latitudes, but it could just as well be described as being an artifact of the curvature of the Earth, and the comparison process.

Different authors take different approaches - Unruh choses to stress the fact that "a second is a second" much as one might chose to stress that "a mile is a mile". But it's also possible to view the situation as distances changing with lattitude.

I'd love to see a picture of the 2-d projection of the 4-d spacetime in the spherically symmetric (but weak) field near a planet. The 2 dimensions would be the radial (or height) and (coordinate-) time dimensions.

Geodesics on that curved 2-d surface - assuming the curvature to be the time curvature only - would show what we perceive as objects falling from large r to small r. What shape is that surface - can it be embedded into a flat 3-d space?

Have you - has anyone here - ever come across a visualization like that? I'm sure it would help me understand the analogy the relationship between gravity and time even better.

While I've seen embedding diagrams of the Schwarzschild geometry, for

instance http://casa.colorado.edu/~ajsh/schwp.html

they usually show r and phi, and suppress t and theta, so they're not what you asked for.

[add]
I think the following *may* work, but I haven't checked it closely. Imagine a sphere,around which you wrap an ace bandage of zero thickness that's infinitely long.

The height of the ace bandage is limited in this analogy - unfortunately you need more than three dimensions to wrap a plane around a sphere. (I think you can do it in 4. Make the bandage too high, and it will intersect itself. But if you have another dimension, you can arrange matters so this doesn't happen.)

Anyway, the finite height of the ace bandage will be a spatial dimension, and the infinitely long dimension of the ace bandage will be the time dimension.

The gaussian curvature of this is positive (there is only one curvature component in 2d) and uniform, so I think it's the right analogy.

 

[MeJennifer]

If this is so - particles make spacetime. Then what makes the gravitational field? I believed large masses were their cause, but small particles make big masses - so it still leaves what makes the gravitational field?

Spacetime is the combined gravitational field of all particles.

 

[intel]

...time and space are emergent properties of the gravitational field...

Can I take this and say that spacetime, space and time are the direct result of particles or any matter which came into existence at t=0, the big bang?

 

[jimbobjames]

@pervect - that's a good example.

Im trying to figure out the equation of a geodesic on a surface like that of the sphere (or more generally a 2-d surface given as a height function z=f(x,y))

Am I correct in saying that I need to:

1) figure out the metric for the surface in question
2) work out the non-zero christoffel symbols
3) write down the geodesic equation(s) and interpret

For the sphere (assume radius 1), I guess the metric is the diagonal matrix with 1 and sine^2 (theta) on the diagonal.

If we were to assume that the equator represents the coordinate time axis, and the zero degree line of longitude were to represent a spatial coordinate, h say (for height), then the metric is diagonal with -1 and sine^2 (t) - right?

I get that only the 0,0,0 and 0,1,1 components of the christoffel symbols are then non-zero - and equal to sin(t)*cos(t) - but the geodesic equation seems a bit strange.

Its like:

dt/(dTau^2) + (sin(t)*cos(t)) [(dt/dTau)^2 + (dh/dTau)^2 = 0

but I may have made a mistake - I've read a bit, but I have'nt practiced until now. The result should be the equation of a great circle - like the equator itself - but I can't see that from this equation.

Any tips?

 

[Mentz114]

$$ds^2 = dr^2 + r^2d\theta^2 +r^2sin^2\theta d\phi^2$$
is the spherical metric for polar coordinates r, theta, phi. If you are restricted to the surface then set dr = 0.

But you want the 4D space

$$ds^2 = dr^2 + r^2d\theta^2 +r^2sin^2\theta d\phi^2 - c^2dt^2$$

 

[jimbobjames]

Thanks Mentz114.

Actually I was looking for the metric of the 2-d space (sorry if I didnt make that clear - not a 3-1 or a 2-1, just a 1-1 space - I know its not very physical but I wanted to see geodesics on 2-d surfaces).

The only relevant coordinates are theta and phi as far as I can tell. I assumed r=1. And then I put theta = t (for this thought experiment - ie the equator represents the coordinate time axis). And I put r*dphi = dphi (r=1) = dh

thats why I got:

ds^2 = - c^2 dt^2 + sin^2 t dh^2

(but do correct me if this is wrong - will check out latex later).

 

[Mentz114]

$$ds^2 = -c^2 dt^2 + sin^2( t )dh^2$$
is what you've written. Which is very weird. I'll work out the 3D problem you stated earlier.

If we were to assume that the equator represents the coordinate time axis, and the zero degree line of longitude were to represent a spatial coordinate, h say (for height), then the metric is diagonal with -1 and sine^2 (t) - right?

I'm not sure what you're doing here.

To get the geodesics on the surface of a sphere you need the metric I posted earlier, with dr=0, r constant.

 

[Mentz114]

For this metric -
$$ds^2 = d\theta^2 +sin^2\theta d\phi^2 - c^2dt^2$$
The Christoffel symbols are -

$$\Gamma^\phi_{\theta\phi} = cot\theta$$

and
$$\Gamma^\theta_{\phi\phi} = -sin\theta cos\theta$$

 

[jimbobjames]

Thanks again.

What I am trying to see are geodesics on 2-d curved surfaces where time is one of those dimensions (and not an extra third dimension). So I am looking at 1 space and 1 time coordinate which I am calling t and h respectively (not 2 space plus 1 time like you have done).

If you put theta = t and phi = h then you get what I have written from what you have written.

$$ds^2 = -c^2 dt^2 + sin^2( t )dh^2$$

(But I am not an expert so now that youve hopefully understood what I am trying to do, can you check this please?)

Coordinate Time (t) along the equator corresponding to theta (because the radius is 1) and spatial height (h) going from the equator to the north pole. 1 space and 1 time dimension.

Cheers.

 

[Mentz114]

Using your last metric I get again -

$$\Gamma^h_{th} = cot(t)$$
$$\Gamma^t_{hh} = -c^{-2}sin(t)cos(t)$$

 

[Mentz114]

Coordinate Time (t) along the equator corresponding to theta (because the radius is 1) and spatial height (h) going from the equator to the north pole. 1 space and 1 time dimension.

Your metric describes a spacetime where space changes direction like a rotating arrow. So a regular movement along h results in a circle.

Very different from the home cosmos.

 

[jimbobjames]

I appreciate it. Ill check where I made a mistake with the christoffel symbols.

"Very different from the home cosmos."

indeed

I was actually considering only a small local piece of the surface to have that curvature - before you get to the north pole, the space dimension would have flattened out again. Same for the t direction (ie before you go back in time it would be flattened out again).

Am also trying to figure out the metric and christoffel symbols for the hyperbolic paraboloid (saddle) described by

z = y^2 - x^2

and for the the sinc function

z = f(r,theta) = sin(r)/r

You seem to be fast at this - any tricks or just lots of practice?

Am I correct in saying that the coordinate just gets a minus sign in front as soon as you say that's the time coordinate?

Its interesting that geodesics on space-space surfaces can go in any direction, but as soon as you replace one space coordinate with time, the geodesic equation restricts you to a subspace of the forward time direction (light "cone"). You get "pushed" forward through time "at the speed of light" when you don't move in any of the spatial directions. And the time curvature curves the free-fall path to lower "heights" - so you could say curved time => gravity.

I suppose to get back to our cosmos I should be just looking at the projection of the 4-space described by the Schwarzschild metric onto the 2-d r,t surface . ie put dtheta and dphi both equal to zero and get:

$$ds^2 = -c^2(1-2M/rc^2) \, dt^2 + \frac{dr^2}{1-2M/rc^2}$$

which in weak fields is I believe approximately:

$$ds^2 = -c^2(1-2M/rc^2) \, dt^2 + dr^2$$

ie it only has time curvature (Edit: as far as I can tell, but I will check this approximation later).

What do you get here for the christoffel symbols?

 

[Chris Hillman]

Unruh's "Slogan" for GTR

To quote from ‘Time, Gravity and Quantum Mechanics, by W. G. Unruh’;
“A more accurate way of summarizing the lessons of General Relativity is that gravity does not cause time to run differently in different places (e.g., faster far from the Earth than near it). Gravity is the unequable flow of time from place to place.

That's a very interesting slogan and after a minute's reflection, I like it! Except for the fact that I fear that most newbies will completely misunderstand it, as I think you have done. The key is that the word "unequable" hints that comparing "elapsed times" measured by different observers is not as straightforward as in Newtonian physics, or even as in flat spacetime.

Unruh's first point is one which I find myself making repeatedly to counter misleading language in many bad popular books (and in forums like this one). His second point, well, before you can grasp that you need to have the correct visual intuition in terms of local light cones and distances taken along two curvilinear arcs sharing the same endpoints in a Lorentzian spacetime--- otherwise you'll probably misunderstand what he's trying to get at here. As a self-test, other readers can try to understand a la Unruh the "gravitational red shifting" of signals sent radially from an observer hovering over a Schwarzschild object to a more distant hovering observer. (Hint: divergence of two initially parallel and nearby null geodesics, both cut by two timelike geodesics, the world lines of the two observers.)

It seems that perhaps gravity and time should be equated

No, no, no! That's certainly not at all what he's trying to suggest!

I am trying to put together a paper using this perspective

For schoolwork? From "perhaps gravity and time should be equated" I think you need to back up and get some geometrical intuition. I think Geroch, General Relativity from A to B is just what you need.

Can I take this and say that spacetime, space and time are the direct result of particles or any matter which came into existence at t=0, the big bang?

Certainly not! Intel and Dravish, I think you both need to be very very careful about drawing conclusions from verbal descriptions. These can play a valuable role, but to understand gtr you'll need to understand the math. That said, I think the book by Geroch does a fabulous job at getting absolutely the most out of the least mathematical background. Also, posters like pervect and robphy are far more reliable sources than some other regular posters here. As I think they will agree, ultimately, good textbooks (and perhaps local faculty, especially if they specialize in gtr) are still your most reliable sources of information.

"jimbobjames" asked about computing the geodesic equations, say for a line element obtained from a "Monge patch". The most efficient way to do this is via the geodesic Lagrangian, which can be read right off the line element. This is well explained in Misner, Thorne, & Wheeler, Gravitation and in other books. I've also explained it in great detail in long ago posts to sci.physics.* which can probably be found via Google (search on "Coordinate Tutorial"). And be careful, you did not correctly compute the linearized Schwarzschild line element correctly and your claim that "it only has time curvature" is incorrect; see MTW. I hope you are not going to join the group of posters who insists on pretending to stating misinformation as fact--- if you don't have good reason to be confident, you should sprinkle your computations with "I believe that", or even restrict yourself to asking pervect questions until you know more.

Here's a simple example. Consider $H^{1,1}$ ("two dimensional de Sitter spacetime") in the lower half plane chart with line element
$$ds^2 = \frac{-dt^2 + dx^2}{t}, \; \; -\infty < t < 0, \; -\infty < x < \infty$$
Read off the geodesic Lagrangian
$$L = \frac{ -\dot{t}^2 + \dot{x}^2}{t}$$
where $u \mapsto \dot{u}$ denotes differentiation with respect to the parameter $s$ of our unknown geodesic. The Euler-Lagrange equations read
$$0 = \frac{d}{dt} \, \frac{\partial L}{\partial \dot{t}} - \frac{\partial L}{\partial t}, \; \; 0 = \frac{d}{dx} \, \frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x}$$
After simplifying, this reduces to
$$\ddot{x} = \frac{2 \, \dot{t} \, \dot{x}}{t}, \; \; \ddot{t} = \frac{ \dot{t}^2 + \dot{x}^2}{t}$$
from which you can read off the Christoffel symbols, although you don't really need these since at this point you already know the geodesic equations in the given chart. Partially integrating the first equation gives $\dot{x} = t^2/B^2$ where $B$ is a constant of integration. Plugging this into the second equation gives
$$t(s) = B \, \sec(s), \; \; x(s) = A + B \, \tan(s)$$
where $A$ is another constant of integration. (Note circular trig functions, not hyperbolic trig functions!) Eliminating the parameter shows that the timelike geodesics can be characterized as hyperbolic arcs which are orthogonal to $t=0$, which represents "timelike infinity" in a Penrose-Carter chart. This chart covers only the "top half" of $H^{1,1}$ if you think of an embedding in $E^{1,2}$ as a hyperboloid of one sheet; the "equator" is at "$t=-\infty$" in this chart.

 

[pervect]

As I understand it, the goal is to embed some slice of the Schwarzschild geometry, considering only the (R,T) coordinates and suppressing Theta and Phi in such a way that this 2-d Lorentzian geometry is embedded on the surface of some 3-d Minkowskian manifold.

This isn't the "standard" embedding, but it seems to me it's a useful one. The "standard" embedding is strictly Euclidiean, this embedding is Lorentzian. The goal is to come up with a curved surface on which we can draw space-time diagrams, and get the correct results.

Ideally we'd like to embed the whole geometry, but the only approach I've come up with will embed only a slice of the geometry.

I think the following will work, though it's unfortunately possible I've screwed up somewhere.

Note that this isn't quite the same as my previous proposal which wasn't fully thought out. The new proposal is a little bit better thought out, but hasn't really been checked very well yet, it could still have fatal errors.

We map (R,T), the Schwarzschild coordinates, into cylindrical coordinates, r, theta, z

Setting rs=1 (or M=1/2) for convenience, we want the resulting metric to be

dR^2 / (1-1/R) - (1-1/R) dT^2

The Minkowskian manifold will be

dz^2 + dr^2 - r dtheta^2

where -pi < theta < pi

Theta is really cyclic, but we restrict it from wrapping around by treating only the interval -pi < theta < pi

This is rather funky (is it really a Minkowskian geometry? I'm not quite positive) but a 2-d surface of this geometry should be a Lorentzian geometry, which is what we want, and the metric coefficients are all +1 or -1.

We can write:

theta = T

thus we are mapping only a finite range of T, -pi to pi. It appears to me we have a choice of what range of T we want to map, a free constant, i.e. we could write T = k theta. For convenience, I set k=1.

we also write

r = r(R)
z = z(R)

It's immediately obvious that r(R) = sqrt(1-1/R) (given our choice of rs=1 and M=1/2 for the Schwarzschild geometry).

z(R) appears to me to exist, but I can only write it down as an integral. Near R=1, it should have some value proportional to sqrt(R-1).

$$z(R) = \int 1/2\,{\frac {\sqrt {R \left( R-1 \right) \left( 2\,{R}^{2}-1 \right) \left( 2\,{R}^{2}+1 \right) }}{{R}^{2} \left( R-1 \right) }} dR$$Far away from the massive body, the geometry is cylindrical, i.e. essentially flat - a cylinder has no intrinsic curvature.

The event horizon maps to a point. (This isn't quite right, as the event horizon isn't really a point, but if we restrict ourself to regions outside the event horizon, I think it's OK).

Restricting T to the range -pi to pi, and also R > 1, it should be possible to draw space-time diagrams on the surface of this geometry and get the proper Lorentzian behavior, allowing one to geometrically visualize "gravitational time dilation", etc.

 

[Chris Hillman]

Go to http://math.ucr.edu/home/baez/RelWWW/ and search the arXiv on "embedding Schwarzschild". There are many papers exploring this kind of pedagogy. I have to say however that in my experience, many posters will be too impatient to acquire correct intuition for $E^{1,2}$ embeddings and thus are likely to misunderstand the gorgeous pics in many of these papers. My advice to newbies to start with Geroch still stands.

 

[pervect]

OK, thanks:

http://arxiv.org/PS_cache/gr-qc/pdf/9806/9806123v3.pdf

looks like the sort of embedding we want.

 

[jimbobjames]

This is great stuff. Thanks guys. Plenty to chew on here for a while.

"And be careful, you did not correctly compute the linearized Schwarzschild line element correctly and your claim that "it only has time curvature" is incorrect; see MTW."

I will check that in MTW. I somehow recall reading where only g00 was significant in weak fields and with non-relativistic mass/speeds. And 2M/rc^2 seems pretty close to zero I would have thought - that's why I was quite confident it was correct. But thanks for pointing that out - I will double check it.

http://arxiv.org/PS_cache/gr-qc/pdf/9806/9806123v3.pdf

This is great - I've listened to Donald Marolf speaking (online videos) - he really is a great teacher too!

 

[Chris Hillman]

Weak-field versus non-relativistic motion

"And be careful, you did not correctly compute the linearized Schwarzschild line element correctly and your claim that "it only has time curvature" is incorrect; see MTW."

I will check that in MTW. I somehow recall reading where only g00 was significant in weak fields and with non-relativistic mass/speeds.

In "the weak-field approximation", you treat the source as first order in some parameter. In particular, you can expand the line element for the Schwarzschild vacuum in a Taylor series about m=0, and then drop all terms O(m^2) and higher. But as you know, $\frac{1}{1-x} = 1+ x + O(x^2)$. Indeed, if you write out
$$ds^2 = -(1-2 \, \phi) \, dt^2 + (1+2 \, \phi) \; \left( dx^2 + dy^2 + dz^2 \right)$$
where $\phi$ is a smooth function of x,y,z only, and expand the Einstein tensor to first order in $|\phi|$, this vanishes provided that the Laplacian of $\phi$ vanishes, so this is a weak-field static vacuum solution, as Einstein himelf pointed out. The point is of course that weak-field gravitostatics in gtr is in this sense formally equivalent to "the Newton-Laplace field theory of gravitation" (an anachronism, but you probably know what I mean).

But don't confuse weak-field solutions with two important but more stringent approximations: slowly time varying weak-field approximate solutions (i.e., a certain class of Lorentzian spacetimes) and the theory of nonrelativistic test particle motion in such spacetimes.

 

[Mentz114]

JimBob,

You seem to be fast at this - any tricks or just lots of practice?

Lots of practice, and some software.

Am I correct in saying that the coordinate just gets a minus sign in front as soon as you say that's the time coordinate?

Space and time parts of the metric have opposite sign so we preserve the hyperbolic Reimann features of SR(?). The convention you choose is called the signature.

I'll leave you to it - I can recommend anything by John Baez.