Can a closed box in freefall reveal the curvature of space?

[russell2pi]

Purpose of the "closed box"

It's often stated that GR follows from the observation that an experimenter inside a closed box in freefall could not distinguish between the box being in that circumstance, and the box being in open space away from any gravitational field. In each case, objects inside the box exhibit inertial motion.

But what if the box had glass walls and the experimenter looked outside? Then it would be obvious -- other free-falling objects outside the box would not appear to exhibit inertial motion.

So what has happened to our ideas of being at rest or constant relative motion? Do they make any sense any more? Does the idea of inertial motion make any sense in anything but a local sense?

How big can the box be and what is it trying to hide? The curvature of space? Is the experiment then telling us, in some sense, that space is locally flat?

Could we somehow define a spatial coordinate system with the box at the origin, with respect to which even distant free-falling objects exhibit inertial motion? I don't see how. Fire two bullets in the same direction at different speeds in space and they will not follow the same path. Tune your coordinates to make one of their paths straight, and the other will appear to accelerate.

Is that where the idea of spacetime curvature comes in?

Is there some way to picture what this curvature looks like, for example around a field created by a point mass? Or is this the point where you can go no further without a mathematical understanding?

 

[pervect]

It's often stated that GR follows from the observation that an experimenter inside a closed box in freefall could not distinguish between the box being in that circumstance, and the box being in open space away from any gravitational field. In each case, objects inside the box exhibit inertial motion.

But what if the box had glass walls and the experimenter looked outside? Then it would be obvious -- other free-falling objects outside the box would not appear to exhibit inertial motion.

So what has happened to our ideas of being at rest or constant relative motion? Do they make any sense any more? Does the idea of inertial motion make any sense in anything but a local sense?

How big can the box be and what is it trying to hide? The curvature of space? Is the experiment then telling us, in some sense, that space is locally flat?

Could we somehow define a spatial coordinate system with the box at the origin, with respect to which even distant free-falling objects exhibit inertial motion? I don't see how. Fire two bullets in the same direction at different speeds in space and they will not follow the same path. Tune your coordinates to make one of their paths straight, and the other will appear to accelerate.

Is that where the idea of spacetime curvature comes in?

Is there some way to picture what this curvature looks like, for example around a field created by a point mass? Or is this the point where you can go no further without a mathematical understanding?

When your box is small enough, you can treat space-time as flat without significant error, just as you can navigate within a small city using flat maps without appreciable error.

For larger things, one needs to use a globe, as the Earth's surface is curved. When you look outside the small box, you run the probability of running into situations where you can't ignore the effects of curvature.

If you know special relativity already, there is a bit further you can go as far as representing the curvature of spacetime. There are several approaches one can use to aid in visulaization, the one I'm most fond of is due to Marolf, http://arxiv.org/abs/gr-qc/9806123.

Basically, if you draw your space-time diagrams on a large surface of the particular shape Marolf computes, you can use the usual Lorentz transforms of SR on small, "nearly flat" sections of the curved surface.

I didn't find the paper particularly easy to read, which i s a pity because the basic ideas are really quite siple.

 

[Dale]

But what if the box had glass walls and the experimenter looked outside? Then it would be obvious -- other free-falling objects outside the box would not appear to exhibit inertial motion.

The point of the box is to restrict yourself to "local" measurements. This does two related things. First, it allows you to approximate the spacetime within the box as being flat, as pervect mentioned. Second, it keeps you from saying that you are accelerating relative to some external object. All you can determine is that you are not undergoing any intrinsic proper acceleration.

This allows you to focus on coordinate-independent things such as the fact that there is no proper acceleration without being distracted by the fact that there is a coordinate acceleration in a reasonable coordinate system like the rest frame of some nearby large gravitating object.

There is no real need for the box, it is just a pedagogical aid to help the student focus on the important things.

 

[HallsofIvy]

It's often stated that GR follows from the observation that an experimenter inside a closed box in freefall could not distinguish between the box being in that circumstance, and the box being in open space away from any gravitational field. In each case, objects inside the box exhibit inertial motion.

But what if the box had glass walls and the experimenter looked outside? Then it would be obvious -- other free-falling objects outside the box would not appear to exhibit inertial motion.

That's not true. Other "free-falling" objects would appear to be motionless relative to the observer. The point of the closed box is that you cannot see objects that are NOT free-falling.

So what has happened to our ideas of being at rest or constant relative motion? Do they make any sense any more? Does the idea of inertial motion make any sense in anything but a local sense?

How big can the box be and what is it trying to hide? The curvature of space? Is the experiment then telling us, in some sense, that space is locally flat?

Could we somehow define a spatial coordinate system with the box at the origin, with respect to which even distant free-falling objects exhibit inertial motion? I don't see how. Fire two bullets in the same direction at different speeds in space and they will not follow the same path. Tune your coordinates to make one of their paths straight, and the other will appear to accelerate.

Is that where the idea of spacetime curvature comes in?

Is there some way to picture what this curvature looks like, for example around a field created by a point mass? Or is this the point where you can go no further without a mathematical understanding?

 

[Naty1]

Does the idea of inertial motion make any sense in anything but a local sense

sure...especially for inertial observers in special relativity...in GR distant inertial motion is likely to be observed as curved...like your two bullets if you omit any frictional effects. And that 'photons follow geodesics' [paths of ideal test particles] is a really, really good approximation, but in fact light of different colors [energies] apparently follows ever so slightly different 'inertial paths' .

Is there some way to picture what this curvature looks like, for example around a field created by a point mass?

[I have to go back and read the Marolf paper...maybe I'll even understand it.] [edit: just looked... oh good grief...not today!]

You are lucky that 3 experts latched onto your question...[not me!]... One thing to keep in mind is that both gravitational potential and acceleration affect 'curvature' ...but the first we technically describe as affecting [gravitational] spacetime curvature, not the second. Others, especially in introductory discussions, don't make the distinction. But you already know that two identical bullets of equal rest mass but traveling at different speeds [same direction] 'curve' differently...follow different paths...so some type of curvature is apparently different between them...You might also be interested to note that if linear bullet velocities are equal but the spins [rotations] are different the bullets will also take different paths.

Anyway, I went through such curvature/path issues several years ago and DrGreg [of these forums] was kind enough to provide a 'picture' for me...and I saved a subsequent closely description, maybe from pervect, on four acceleration:

[I posted this description perhaps a half dzen times and no experts have critiqued nor used an analogous description, so I don't think it's popular, but nobody has called me an idiot either for latching onto it.]

(from Dr. Greg) (my boldface)

"... let's restrict our attention to 2D spacetime, i.e. 1 space dimension and 1 time dimension, i.e. motion along a straight line. …
In the absence of gravitation, an inertial frame corresponds to a flat sheet of graph paper with a square grid. If we switch to a different inertial frame we "rotate" to a different square grid, but it is the same flat sheet of paper. (The words "rotation" and "square" here are relative to the Minkowski geometry of spacetime, which doesn't look quite like rotation to our Euclidean eyes, but nevertheless it preserves the Minkowski equivalents of "length" (spacetime interval) and "angle" (rapidity).)

If we switch to a non-inertial frame ([an accelerated observer] but still in the absence of gravitation), we are now drawing a curved grid, but still on the same flat sheet of paper. Thus, relative to a non-inertial observer, an inertial object seems to follow a curved trajectory through spacetime, but this is due to the curvature of the grid lines, not the curvature of the paper which is still flat.

When we introduce gravitation, the paper itself becomes curved. (I am talking now of the sort of curvature that cannot be "flattened" without distortion. The curvature of a cylinder or cone doesn't count as "curvature" in this sense.) Now we find that it is impossible to draw a square grid to cover the whole of the curved surface. The best we can do is draw a grid that is approximately square over a small region, but which is forced to either curve or stretch or squash at larger distances. This grid defines a local inertial frame, where it is square, but that same frame cannot be inertial across the whole of spacetime.

So, to summarize, "spacetime curvature" refers to the curvature of the graph paper, regardless of observer, whereas visible curvature in space is related to the distorted, non-square grid lines drawn on the graph paper, and depends on the choice of observer..."When we talk of curvature in spacetime (either curvature of a worldline, or curvature of spacetime itself) we don't mean the kind of curves that result from using a non-inertial coordinate system, i.e, non-square graph paper in my analogy.

Four acceleration: two mathematical descriptions

In Minkowski coordinates in Special Relativity, 4-acceleration is just the coordinate derivative of 4-velocity with respect to arc-length (proper time), and the 4-velocity is the unit tangent vector of worldline. As the 4-velocity has a constant length its derivative must be orthogonal to it. The 4-acceleration is the curvature vector; orthogonal to the worldline and its length is the reciprocal of the worldline's "radius of curvature".

In non-inertial coordinates in GR, the 4-acceleration is defined as a covariant derivative. This takes into account (and removes) any curvature of spacetime or "apparent curvature" due to using a "non-square grid", and leaves us with curvature that is a property of the worldline itself, not the spacetime or the choice of coordinates.

Is THIS a valid perspective:

No one has ever explained the stress energy tensor[SET] quite this way to me...but I think it's valid: The SET is the source of gravitational [spacetime] curvature. [no issue there] If you sit in the frame of the object, move along with it and observe any effects of stress, energy, momentum, etc, then that gets attributed to gravitational spacetime curvature. So the velocity of each of your bullets falls outside the frame of the bullet, so we say that's a curvature outside of gravitational effects. [drawing a curved grid on the flat spacetime paper of DrGreg] But the rotational energy of a bullet IS apparent within the frame of the moving bullet, so we say that IS a gravitational effect. [So the graph paper itself becomes curved] [experts??]

 

[PeterDonis]

light of different colors [energies] apparently follows ever so slightly different 'inertial paths'

Do you have a reference for this?

 

[Naty1]

Do you have a reference for this?

Hi Peter...Was the subject of a long and tortuous discusson in these forums...I'll see if I can find the source. I think the point was photons don't behave exactly as ideal 'test particles'...I have posted such several times hoping someone would comment...

I said 'apparently' because I have not seen it elsewhere...like the 'fact' that supposedly two parallel laser beams don't attract when traveling in the same direction, but supposedly do when traveling in opposite directions...haven't figured either out for sure...

 

[Naty1]

light of different colors [energies] apparently follows ever so slightly different 'inertial paths'

Do you have a reference for this?

I found the quote in my notes, but not any source [I only quote from among maybe a dozen people here, so be careful...I could have gotten it from you [lol]:

How does gravity alter the trajectory of light?

“According to Einstein's General Relativity Theory, light will be curved as is mass. All objects with mass or energy alter the curvature of spacetime, the 4 dimensional fabric of the universe. Free falling objects moving through spacetime then simply follow the curves that have been created. As an approximation, we say light follows a null geodesic, so that all color light approximately follows the same geodesic.


You can see the 'logic' used, but am unsure if it is CORRECT logic!

 

[PeterDonis]

I found the quote in my notes, but not any source [I only quote from among maybe a dozen people here, so be careful...I could have gotten it from you [lol]

I don't think that quote was from me.

If we leave out gravity, the only reason why light of different colors would follow different worldlines would be if the photon had nonzero rest mass, so photons of different energies would move at different speeds (i.e., "dispersion"). If that was where the "approximately" came from in that quote, then it was at best misleading, since photon dispersion, if it exists (our best estimate on current data is that it does not, the photon's rest mass is zero), has nothing to do with gravity.

If we now include gravity, and assume the photon's rest mass is zero, the only other thing that I can see that the "approximately" could have been referring to is some possibility that light could travel along a null curve that wasn't a geodesic. That is, perhaps photons, even though they are massless, have some sort of internal "structure" (i.e., they aren't ideal "test particles"), that varies with the photon's energy, and that structure somehow interacts with spacetime curvature to make photons of different energies follow different null curves, even if they are emitted in the same direction. Of course all but one (at best) of these curves can't be geodesics, since from any event there is only one null geodesic in a given direction. So in this case the photons would not be following "inertial paths" (except for possibly one of some specific energy that does move on the null geodesic). I have never seen any serious speculation along these lines, but perhaps someone else has.

Bottom line: if photons have zero rest mass (which they do to the best of our knowledge), I think it is overwhelmingly likely that they all move on null geodesics, so that all photons emitted in a given direction, regardless of color (energy), would move on the same path.

 

[PeterDonis]

the 'fact' that supposedly two parallel laser beams don't attract when traveling in the same direction, but supposedly do when traveling in opposite directions...haven't figured either out either for sure...

This has to do with how electromagnetic fields act as sources of gravity, i.e., what kinds of solutions to the EFE they create and how the solutions for two light beams combine, and how the combination depends on whether the beams are parallel or antiparallel (which it does). But in all these models, AFAIK, individual photons are still treated as zero rest mass "test particles" that move on null geodesics of whatever overall spacetime is determined by solving the EFE with the given sources. So the parallel vs. antiparallel laser beams are a separate question from the behavior of individual photons as "test particles".

 

[Naty1]

if photons have zero rest mass (which they do to the best of our knowledge), I think it is overwhelmingly likely that they all move on null geodesics, so that all photons emitted in a given direction, regardless of color (energy), would move on the same path.

I'm not surprised, because when I came across the statement I posted I wondered why one never reads about a 'rainbow' effect from gravitational lensing...and I concluded well maybe the effect is so small as to be unobservable...anyway thanks for the discussion...

 

[russell2pi]

Second, it keeps you from saying that you are accelerating relative to some external object. All you can determine is that you are not undergoing any intrinsic proper acceleration.

Bang, I think you have hit the nail on the head of what got me thinking about this in the first place.

The idea of "acceleration" is now radically different to the classical concept. It's not a kinematic concept at all... nothing to do with a nonzero second derivative of displacement with respect to an inertial observer. That can value be nonzero even between two distant inertial frames (hence, both with zero proper acceleration).

That's going to take some getting my head around.

That's not true. Other "free-falling" objects would appear to be motionless relative to the observer

How can that be? Consider a second free falling object on the opposite side of the gravitating mass. It would not be motionless. It would accelerating straight for you. Or, consider orbiting bodies at different radii. They are all in relative motion with each other.

 

[Naty1]

How can that be?

the elevator analogy is a limited one; your examples go beyond the intent of the analogy...
so you are correct and so is HallsofIvy.