How does GR handle metric transition for a spherical mass shell?

PeterDonis

If the triangle is *assembled* in the curved space, in the right way, this will be true. But you specified that the triangle (or tetrahedron) was assembled in flat space, and then *moved* to curved space. That means the "natural" configuration of the sides is the flat configuration, Euclidean straight lines, and they must be deformed to get into the curved configuration, geodesics on a curved surface.

If, instead, you assembled the triangle carefully in a curved region, so that the "natural" configuration of its sides was as geodesics on the curved surface, and the "natural" angles at each vertex summed to more than 180 degrees, then the triangle would deform if you tried to make it conform to a flat Euclidean plane. Similarly, if you assembled a tetrahedron in a region of curved space, hovering over a gravitating body, so that it was unstressed in that configuration, and then took it far away from gravitating bodies where space was flat, it would undergo stress and deformation in the course of conforming to the flat space.

It depends on what you want to make an analogy to. If we are trying to construct an analogy to what happens when we take an object constructed far away from gravitating bodies and move it into a gravity well, the above is *not* a good analogy for that, because deflating the balloon corresponds to contracting the spacetime as a whole. That would do as an analogy for a collapsing universe, but *not* for moving into a gravity well. A better analogy for that would be to consider a surface like the Flamm paraboloid...

http://en.wikipedia.org/wiki/File:Fl...paraboloid.svg

...and think of moving a triangle from the flat region to the curved region.

PeterDonis

Then I'm not sure I would call this a "shell". "Shell" implies a thin region of matter with complete vacuum inside and outside it, at least in the context of this thread.

How are you calculating the force?

JDoolin

This is a good question, or more specifically, how should I (or how should we) calculate the force? So far, essentially, I'm not calculating the force. Invoking symmetry, there is no net force. There's a gravitational potential, but no net force on a particle at or near the center of the distribution.

But let's look at the other extreme now, from the very edge of the distribution.


Here, obviously we don't have symmetry. The stationary particle should experience a net force to the right.

Let me give you a couple of premises of how I would go about calculating the force, and see if you agree with these premises, or if I am hopelessly naive.

#1 The net gravitational force on a particle is [tex]F = G m \sum_i \frac{M_i}{r_i^2} \vec u_i[/tex]
#2 The force on a particle should be calculated based on the reference frame of the particle that is undergoing the force.
#3 The mass of each particle affecting gravitation is the rest-mass of the particle. i.e. the Lorentz factor affects momentum, but not gravitational attraction.
#4 The location of the particle is not the "simultaneous" location, but rather the speed-of-light delayed location of the particle. i.e. the speed of gravity is the same as the speed of light, so we must find an intersection of the past-light-cone of our particle of interest with the world-lines of the particles involved.

Another peculiarity of relativity, (whether using Galilean Transformation or Lorentz Transformation) is that when one accelerates toward a future event, it leans toward him, becoming directly in his future, but when one accelerates toward a past event, it leans away. We may find that even if the particles at the edge accelerate "toward" the center, that in fact, the end result is not at all what common-sense would suggest. By accelerating toward the center, the particle continually enters reference frames where the center is further and further away.

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PeterDonis

Well, the problem as a whole has spherical symmetry--or at least, you can impose that condition as a reasonable idealization to make the problem tractable. If the problem has spherical symmetry, then the "force" on any particle (a) must point in the radial direction, and (b) must be a function only of its radial coordinate. If the matter is all ordinary matter, as I described in my last post, then (a) can be further refined: the force on any particle must point radially *inward*, i.e., the expansion of the shell must be decelerating. As I note below, viewing this deceleration as caused by a "force" may not be the simplest way to view this problem.

This requires the problem to be non-relativistic. That is not consistent with your #4. In fact, the inconsistency between #1 and #4 was one of the stumbling blocks on the road to General Relativity, back in the early 1900's. See further note below.

A better way to state this would be: the 4-force on a particle is a covariant geometric object, it is the particle's rest mass times the particle's 4-acceleration, which is the covariant derivative of its 4-velocity with respect to its proper time.

A better way to state this would be: the "gravitational field" is determined by the stress-energy tensor of the matter, which is a covariant geometric object. The SET determines the field via the Einstein Field Equation. It also correctly accounts for the fact that "rest mass" is what affects gravity, not momentum due purely to kinematics.

If you are assuming this, then you can't use #1 as your force equation. Consider a simple example: the force on the Earth at a given instant does *not* point towards where the Sun was 8 minutes ago by the Earth's clock. It points towards where the Sun is "now" by the Earth's clock. (Technically, there are some small correction factors, but they can be ignored for this discussion.) So if you use your #1, you have to plug in the position of the Earth relative to the Sun "now", not the "retarded" position, or you'll get the wrong answer. (Steve Carlip wrote an excellent paper some time ago that explains all this: it's at http://arxiv.org/abs/gr-qc/9909087.)

As I noted above, using #1 requires the problem to be non-relativistic, and you don't seem to be imposing that limitation. For the relativistic case, you can't really use a "force" equation for this problem, or at least it does not seem to be the easiest way to approach it. A better model would be an expanding matter-dominated FRW-type model, as I mentioned in a previous post, especially since it doesn't seem like your particles are a "shell", since there's no interior vacuum region, as far as I can see. This type of model does not view gravity as a "force"; it just solves for the dynamics of a curved spacetime using the EFE and an expression for the stress-energy tensor of the matter. You can still view individual particles as being subject to a "force", but that force can be more easily calculated *after* you have constructed the model from the EFE.

If the spatial extent of the expanding "shell" is limited, then at the surface of the shell, the FRW-type solution would be matched to an exterior Schwarzschild vacuum solution; this would basically be the time reverse of the Oppenheimer-Snyder solution for the gravitational collapse of a star.

Huh? I don't understand what you're getting at here at all. How can you accelerate toward a past event? For that matter, how can you accelerate toward a future event? You accelerate toward a point in space, not an event in spacetime. Also, what kind of "reference frames" are you talking about? If you're talking about ordinary Lorentz frames, they are only valid locally; you can't use them to correctly evaluate distances to faraway objects.

It looks like this discussion might be better moved to a new thread, since it appears to be getting further from the topic of this one.

JDoolin

Okay, I see where I had one mistake.

I was thinking that coulomb's law and gauss law were equivalent. But they are only equivalent when the charge is not moving. I'll try to read more of the article to see what I've missed. May take me a bit of time to catch up so I don't think I'm ready to start another thread at this time.

If indeed, the gravitational field from a receding object has some predictable variation based on its relative velocity, then we should, of course, use that modification. But if you want to claim that there is no predictable variation; no predictable "meaningful" position we can use for distant objects, I beg to differ.

On another topic, if possible, can you support your argument that "ordinary Lorentz frames...are valid only locally." I've heard this time and time again, but no one has ever explained what it means. Are you saying that when I point at a distant galaxy, that that direction that I am pointing only exists locally? Are you saying that the very concept of direction is only a local phenomenon?

PeterDonis

The answer to this will be easier to understand if I answer your other question first. See below.

It means that you can set up a Lorentz frame centered around any event you like in spacetime, but the frame will only obey the laws of special relativity in a restricted local region of spacetime around that event. Strictly speaking, it will only obey the laws of special relativity exactly *at* that event; but in practice, there will be a finite region around the event where the deviations from the laws of special relativity due to the curvature of spacetime are too small to detect. The size of that region depends on how accurate your measurements are and how curved the spacetime is.

Here's a simple example: take an object that is freely falling towards the Earth from far away (say halfway between the Earth and the Moon), and set up a Lorentz frame using its worldline as the t axis. Since the object is in free fall, i.e., inertial motion, its worldline can be used this way according to the laws of SR. Now take a second object which is also in free fall towards the Earth, but which is slightly lower than the first object. Suppose there is an instant of time at which the second object is at rest relative to the first; we take this instant of time to define t = 0 in our Lorentz frame, and the position of the first object at this instant to define the spatial origin, so the frame is centered on that event on the first object's worldline.

It should be evident that, for a given accuracy in measuring the relative velocity of the two objects, there will be some time t > 0 at which it becomes apparent that the second object is no longer at rest relative to the first. (This is because it is slightly closer to the Earth and therefore sees a slightly higher acceleration due to gravity.) But SR says that' can't happen: both objects are moving inertially, both were at rest relative to each other at one instant, and SR says that they therefore should remain at rest relative to each other forever. They don't. So we can only set up a Lorentz frame around the first object's worldline for a short enough period of time that the effects of tidal gravity (which is what causes the difference in acceleration of the two objects) can't be measured. Similarly, if the second object were further away from the first, its relative acceleration would become evident sooner; so there is a limit to how far we can extend a Lorentz frame around the first object in space as well, before the effects of spacetime curvature become evident.

The direction you are pointing is the direction from which light rays emitted by the distant galaxy are entering your eyes. It is certainly reasonable to call that "the direction of the galaxy", but only if you are aware of the limitations of that way of thinking. Light paths are bent by gravity, so the direction you are seeing the light come from may not be "the" direction the galaxy is "actually" in. This is another manifestation of spacetime curvature, and it means you can only set up a Lorentz frame locally with regard to directions and positions as well as relative motion of objects.

This also applies to what you were saying about assigning a "distance" to distant objects. There is no unique definition of "distance" in a curved spacetime; there are at least three different ones that are used in cosmology. You can uniquely define distance in a local Lorentz frame, but that only works locally; once you are out of the local region where the laws of SR can be applied to the desired accuracy, your unique definition of distance no longer works.

pervect

There's a well known solution for an expanding (or collapsing) sphere of pressureless dust. Which isn't quite what was asked for, but you might be able to graft onto it to get the solution. The expanding pressureles dust sphere solution is just the FRW solution of cosmology, by the way.

As far as forces go, they're pretty much not used in GR. You can calculate them as an afterthought when you have the metric by evaluating [itex]u^a \nabla_a u^a[/itex], where u^a is your velocity vector.

[add]Conceptually, what the above expression does is calculate what an accelerometer following the worldline would read.

The reason forces aren't used much in GR that they transform in a complex manner - i.e. they don't transform as a tensor.

In SR, you can still deal with 4-forces as a tensor under Lorentz boosts. In GR, with accelerated coordinate systems, forces obviously cannot transform as a tensor. For instance, if you have an unaccelerated system with zero force, an accelerated system will have a nonzero force, but a tensor that is zero in one coordinate system must be zero in all.

Q-reeus

No problem in agreeing with that situation.
In that more generaized context I see the point, no problems.

Q-reeus

Handy reminder; while I think DrGreg had posted on it much earlier, upon finding the original article at http://en.wikipedia.org/wiki/Flamm%2...27s_paraboloid , studied that piece with some more attention. Had never bothered to understand the significance of the w ordinate, but now appreciate how it gives a useful handle on visualising spatial part of SM. 'Inverting' w ordinate gives the sense of radial contraction as I had envisaged. Matching that to shell transition is the interesting exercise. Brings it back to on topic, and having come to appreciate the general view of the difficulties of applying a simple reading of SC's, conclude the original posting was an inappropriate vehicle for finding problems with SM. Thanks for all the inputs.

JDoolin

I think my main objection to this is that if two objects are in free-fall, you cannot claim that they are both moving inertially.

Although, depending on what you mean by "set up a Lorentz frame using its worldline as the t axis," I may have an even greater objection:

Are you suggesting that geodesic paths in a gravitational field can represent straight lines in global inertial Lorentz Frames (in which case, I cannot agree; curved paths are not straight lines) or are you saying that a momentarily comoving reference frame with time axis tangent to the geodesic worldline at one particular event represents an inertial Lorentz Frame?

In any case, this no longer has anything to do with shells, so I can invite you to read my comments here: http://www.physicsforums.com/showthread.php?t=545002 or if you prefer, you or I could start a new thread regarding whether Lorentz Transformations are purely a local phenomenon.

PeterDonis

Um, excuse me? "Moving inertially" and "in free fall" are the same thing.

Obviously not; the whole point is that geodesic paths in a gravitational field are not straight lines in a global Lorentz frame. That's why SR cannot be applied on a large scale in a curved spacetime.

Yes, that's a more precise way of stating what I meant by "use the worldline of one freely falling body as the t axis of the Lorentz frame". So it seems that you understand that I can only set up a momentarily comoving reference frame tangent to a worldline in a curved spacetime at one particular event. That's what "Lorentz frames are only valid locally" means. So I'm having trouble understanding why you're not clear about the meaning of "Lorentz frames are only valid locally".

I'll look at what you posted in the other thread and make further comments on that if need be.

PeterDonis

Quick answers to the questions you posed in the other thread; I won't comment in that thread unless it seems warranted.

(1) my answer is (b)

(2) my answer could be (a), but only in the vacuous sense; in the presence of gravity there are no flat regions of spacetime, there is some curvature everywhere. (b) is the strictly correct answer, but as I said in an earlier post, for a given accuracy of measurement there will be some finite region where the deviations from flatness are not observable, and within that region a Lorentz transformation on a local coordinate patch of "flat" Minkowski coordinates will work

(3) my answer is (b), but "similar" has to be interpreted carefully; Fredrik's comments in an earlier post in that thread on the properties a coordinate chart has to have to admit these transformations are worth reading

(4) my answer is "it depends"; (a) is correct *only* if the coordinate chart meets the requirements for having a rotation transformation in the first place, per Fredrik's post; otherwise the answer is undefined because there is no well-defined assignment of coordinates to objects beyond a small local coordinate patch; (b) is never correct, if the transformation is valid at all when extended to distant objects it will change their coordinate positions

Passionflower

As Peter already wrote, inertial movement and free fall refers to the same thing.

One can have many objects in free fall all with lower and higher local velocities wrt a free falling object at escape velocity.

JDoolin

Newton's first law is often referred to as the law of inertia. The velocity of a body remains constant unless the body is acted upon by an external force.

An object in free fall is changing its velocity because it is acted upon by an external force. The force of gravity.

Inertial movement is movement in a straight line.

Free-fall is not inertial movement. It's accelerated movement.

This is a basic reality-check. You mean to tell me that if an object is in orbit, you actually consider that to be a straight line?

PeterDonis

Ah, ok. We have a terminology problem. See below.

Not in GR. In SR, yes, but that's only because SR assumes spacetime is flat. In GR, where spacetime can be curved, inertial motion has to be defined physically. An object is moving inertially if it feels no force, i.e., is weightless, i.e., is in free fall.

In Newtonian terms, yes. But that's because Newtonian physics defines "force" and "acceleration" differently than GR does. In GR, an object in free fall, that is weightless, *feels* no force, and therefore there is no force on it in any coordinate-independent, invariant sense, and it is *not* accelerated. More precisely, it is not accelerated in the coordinate-independent sense of "acceleration": the covariant derivative of its 4-velocity with respect to its proper time is zero. This is sometimes called proper acceleration or 4-acceleration. And since "force" in GR is defined as the object's rest mass times its proper acceleration, a freely falling object is experiencing zero force in GR.

The object is "accelerated" with respect to the Earth, but that is coordinate acceleration, not proper acceleration; similarly, the "force" of gravity in Newtonian terms is not a "force" in GR, because it does not cause any 4-acceleration of the object. That's how I was using the terms, and how they are standardly used in GR.

In GR, yes. More precisely, the object's *worldline* is a geodesic of the curved spacetime around the Earth, so it's as "straight" as any worldline can be in that spacetime. I was not saying that the object's path in *space* was straight; obviously it's not. But that's irrelevant to this discussion.

Passionflower

It is not really relevant to the question what is free falling and inertial, every test observer that has no proper acceleration is free falling and inertial.

From one perspective one certainly could consider it a straight line as a test object in orbit differs only from a radially free falling test object by having a non-zero angular momentum. For instance do you feel yourself turned often when the seasons pass? The same for an astronaut who travels in orbit, as far as he is concerned he travels in a straight line as no forces act on him.

Passionflower

I disagree with that.

Whether a free falling object's path in space is straight depends solely on the chosen coordinates. In simple terms, the Sun just as much goes around the Earth as the Earth goes around the Sun it simply depends on the point of view.