Exploring the Effects of Relative Mass and Relativity on Gravitational Pull

[pervect]

The metric I gave is an exact solution. It is simply the Schwarzschild solution written in funny coordinates. All I did, after all, was take the Schwarzschild solution and do some coordinate changes.

The funny coordinates happen to asymptotically approach the coordinates of an inertial observer at infinity boosted with respect to the black hole.

If you want to know about inertial observers at a finite distance from the black hole, you have more work to do. You need to find geodesics for "flyby" orbits that come in from infinity, approach to a minimum distance b, and then leave toward infinity. This is not hard to do...in fact, it might be easier in the original Schwarzschild coordinates.

If I remember correctly, the whole point of this thread was to try to make sense of "relativistic mass" in a GR context...and I think we have all shown in various ways that it really doesn't make any sense. GR doesn't work that way.

The metric is exact before the boost - but the boosting process is an approximation, I think, that works "far enough away". Meanwhile I'm looking for typos and whatnot...

 

[Ben Niehoff]

The "boosting process" is just a coordinate change. Einstein's equations are generally covariant, which means that they continue to hold under any coordinate changes. Hence the boosted metric is still an exact vacuum solution. There is no trickery here.

But like I said, there's nothing special about the new coordinates either. To find out what physics are experienced by observers at a finite distance, you need to compute geodesics and look at local inertial frames, in either metric.

 

[pervect]

By the way, they won't, because

R_{\mu\nu\lambda\rho} u^{\nu} u^{\rho}

is a coordinate-dependent object (it still has free indices). It also doesn't represent tidal forces! Forces, like all physical quantities, need to be measured in a local inertial frame.

I don't have a basis-vector form for the metric you proposed ,but if you calculate the above expression in a local frame-field with basis vectors that are Minkowskian, it does have a physical interpretation as the tidal forces along the chosen axes.

Meanwhile the metric coefficients at large R should be close to being diagonal, meaning that the coordinate vectors should be almost Minkowskian, but that's another approximation.

See for instance MTW pg 860, for where they use something similar to the above to calculate the tidal forces on an observer falling into a black hole.

It's sometimes called the tidal tensor, or the electrogravitic tensor. You're right in that it does need to be measured in an inertial frame to be interpreted as a tidal force - if the frame is rotating, for instance, the centrifugal forces will interefere with the measurement. That shouldn't be an issue here, the observer in this case is in free fall, though rotation of our observer relative to the fixed stars is an issue

 

[PAllen]

I though of another way to describe describe the procedure I have in mind from post #18.


Imagine some inertial observer (at rest in their frame) very far from rapidly moving massive body - far enough that spacetime is Minkowski flat to any desired precision. Now assume they extend a very long, negligible mass, ruler of given proper length toward the trajectory to be followed by the massive body. Assume they are able to anchor this ruler so it doesn't move at all from their point of view. At the end close to where the massive body will approach, they have an accelerometer. As the massive body whizzes near this ruler, the accelerometer reading will vary in direction and magnitude, with some point of maximum magnitude. That maximum magnitude is my measure of gravitational force of passing massive body at effective closest approach.

What we have produced here is a distant reference world line that is Minkowski straight, another world line 'close to the action' defined in reference to it (the test body world line), and a measure that corresponds to maximum proper acceleration along the test body world line. Having defined this test body world line, if we transform to coordinates where the massive body is stationary, and compute maximum proper acceleration of the test body world line in these coordinates, we must necessarily get exactly the same value.

Thus I have defined a physically plausible, invariant measure of 'gravitational attraction' between a massive body and a test body in rapid relative motion. Guaranteed to come out the same in whichever coordinates it is analyzed in.

 

[pervect]

The "boosting process" is just a coordinate change. Einstein's equations are generally covariant, which means that they continue to hold under any coordinate changes. Hence the boosted metric is still an exact vacuum solution. There is no trickery here.

You're right about that...

 

[pervect]

If I take the limit as x-> infinity, by considering only the highest order terms in x (the only way to deal with the very messy expressions), I'm finally getting the same answers for the tidal forces.

I think some comment on the case on the error induced in tidal forces from acceleration is useful - not so much for the moving case, as for it's application to the static case. If one is accelerating at 1g, assuming I've done the numbers right, one will falsely measure a stretching tidal force of about 10^-16 g / meter in the direction of acceleration (an error of 1+gz/c^2).

This measurement is done just by mounting accelerometers on "rigid rods" in the fermi-normal frame of the accelerating observer, and comparing the accelerometer readings. The accelerometer "above" the observer will be slightly low, the one "below" slightly high, hence an apparent stretching force.

So it's not a big issue in practice with reasonable accelerations.

 

[TWest]

You all have been debating for days. So any progress on the subject have you found a acceptable way to explain it without much error?

 

[PAllen]

You all have been debating for days. So any progress on the subject have you found a acceptable way to explain it without much error?

Not really, and I'll explain my view of why you may never get a clear, consensus answer.

In Newtonian physics, gravity is a force dependent only on mass and distance (not relative motion), and mass and distance are well defined, simple quantities. In GR, gravity is not a force, there is no unambiguous definition of mass at all (really), and distance is not simple when large. In the case of slow motion, and solar system scales, these complications wash away, and a clear answer can be given. However, for relativistic motion between a massive body and e.g. a rocket, there is no unambiguous answer. In my view, what one would like to answer is:

Imagine a rocket moving rapidly, inertially, against the local stars, far from any massive body. Then imagine it approaching and passing near a massive body. How much force will the rocket's thrusters need to exert to maintain its original path? The problem is, that since gravity in GR manifests as the geometry of spacetime, and the path you want to maintain has no unique geometric features (i.e. not a geodesic), there is no unambiguous answer to 'maintain original path'. Each of Bill_k, Pervect, and I have proposed different ways around this ambiguity, and I don't think you can say that any of them are right or wrong. Ben Niehoff has displayed a metric which would make it conceptually easy to implement my proposal, but it would still be a substantial computation even with the aid of symbolic software. As a result, my proposal will remain uncomputed.

 

[Dale]

You all have been debating for days. So any progress on the subject have you found a acceptable way to explain it without much error?

What is clear, however, is that you cannot simply take Newtonian gravity and substitute relativistic mass.

 

[atyy]

You all have been debating for days. So any progress on the subject have you found a acceptable way to explain it without much error?

Bill K gave the answer very early in the thread (post #7).

 

[PAllen]

Bill K gave the answer very early in the thread (post #7).

Well, Bill_k said his approach has the feature that it would come out different in different coordinates. I have a problem with that for something considered 'observable'.

 

[atyy]

Well, Bill_k said his approach has the feature that it would come out different in different coordinates. I have a problem with that for something considered 'observable'.

The question is ill-specified because we don't know what 0.99c is relative to. In plain English, I'd say its the relative velocity of the moon and earth, which of course doesn't exist in GR. So to make the question answerable, coordinates (or maybe even a local frame) must be chosen. Bill K interpreted the question so that it makes sense, and answered it. DaleSpam interpreted the question differently, and also answered it (without further specification, mass is not defined in GR). Both seem to me correct answers to their interpretations of the question.

But you have a point. The usual presentation of the Shapiro time delay is unobservable. Its verification involved quite a few additional steps.

 

[Naty1]

I take this whole discussion to be summarized by post 39 and 40...I hope!

What I think Dalespam is saying is that the original post computation is incomplete...GR is a tensor based phenomena...not a simple mass adjustment of Newtonian expressions.

Gravitational attraction IS based on mass, energy, pressure...the "stress energy momentum" tensor...so anything that increases any components in that expression increases the gravitational field.

This same issue has come up in at least one very recent discussion in which no "experts" participated...and in which there was lots of confusion judging by the posts there.

 

[PAllen]

The question is ill-specified because we don't know what 0.99c is relative to. In plain English, I'd say its the relative velocity of the moon and earth, which of course doesn't exist in GR. So to make the question answerable, coordinates (or maybe even a local frame) must be chosen. Bill K interpreted the question so that it makes sense, and answered it. DaleSpam interpreted the question differently, and also answered it (without further specification, mass is not defined in GR). Both seem to me correct answers to their interpretations of the question.

But you have a point. The usual presentation of the Shapiro time delay is unobservable. Its verification involved quite a few additional steps.

To make the question answerable and represent an observable (which must be coordinate invariant), you need to specify only two things: the background geometry, which all have taken to be spherically symmetric, static solution; and the world line of an accelerometer (which would involve e.g. .99c speed relative to the massive body). It is true that there is no unique prescription for the accelerometer world line. It is true that this is computationally difficult. However, for any choice of accelerometer world line, this procedure gives an invariant scalar answer. Any measurement in the real world would be a realization of this exact procedure.
So, we do not really have consensus on what constitutes a reasonable answer to the question.

[EDIT: My intuition suggests that for a reasonable choice of accelerometer world line, we would get the same gamma^2 factor as Bill_K got, but now as coordinate invariant scalar. However, I have not been able to derive this in a credible way.]

 

[atyy]

To make the question answerable and represent an observable (which must be coordinate invariant), you need to specify only two things: the background geometry, which all have taken to be spherically symmetric, static solution; and the world line of an accelerometer (which would involve e.g. .99c speed relative to the massive body). It is true that there is no unique prescription for the accelerometer world line. It is true that this is computationally difficult. However, for any choice of accelerometer world line, this procedure gives an invariant scalar answer. Any measurement in the real world would be a realization of this exact procedure.



So, we do not really have consensus on what constitutes a reasonable answer to the question.

[EDIT: My intuition suggests the for a reasonable choice of accelerometer world line, we would get the same gamma^2 factor as Bill_K got, but now as coordinate invariant scalar. However, I have not been able to derive this in a credible way.]

Well, why don't we take a leaf from Shapiro, ie. compare to the Newtonian case, which is very much how Bill K approached it. You will object there's no Newtonian case to compare to, so it is not observable. But this can be gotten round in the Shapiro case by being a bit more careful, and I would guess one could do it for Bill K's answer too. Maybe we can look at some PPN parameter?

 

[pervect]

Actually, I think Bill's approach gives the right order of magnitude - if you're going to try to think in terms of forces, thinking of the force as going up by one factor of gamma due to the increased energy of the source, and another factor of gamma due to the field being squished, gives semi-reasonable results in most cases. Though I think I'd lose about half or more of the audience when I talk about the "field being squished". Probably because it's not terribly clear what that means, though I'm not sure how to explain it better without spending a lot of words. It's similar to the way the electric field gets squished in this diagram:

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_15.pdf

Maybe I should just draw a similar diagram, I don't think I can find better words than "the field gets squished and is no longer spherically symmetrical".

In the E field case we have the "same number of flux lines", that's not the case for gravity, if we had flux lines, there'd be a factor of gamma more, so I'd need to modify the diagram slightly (but gravity isn't a 2-form like E&M, so the idea of flux lines isn't the powerful tool for gravity it is for E&M, it might be clearer just not to mention them).Now, I don't know what to do about cautions - other than to say that you can expect up to 2:1 (maybe even higher?) errors if you take this idea too seriously and try to actually compute things with it - for instance, if you try to use this idea of "force" to compute trajectories, or tidal forces, or the gravitational just inside the surface of a shell containing a relativistic gas, or the amount of induced velocity after a relativistic flyby (ala Olson, et al, "Measuring the active gravitational mass of a moving object".) Or pretty much anything, actually...

 

[atyy]

http://books.google.com/books?id=w4Gigq3tY1kC&source=gbs_navlinks_s" for GR.

Does that settle it It's in the spirit of Bill K's calculation, though I am not sure if the Minkowski+perturbation split is the same one he chose.

 

[pervect]

You all have been debating for days. So any progress on the subject have you found a acceptable way to explain it without much error?

IS 2:1error "not much?"?

 

[pervect]

http://books.google.com/books?id=w4Gigq3tY1kC&source=gbs_navlinks_s" for GR.

Does that settle it It's in the spirit of Bill K's calculation, though I am not sure if the Minkowski+perturbation split is the same one he chose.

IF I recall that quote comes from the PPN section? PPN is powerful, but I believe it makes some specific assumptions about the source, specifically about the rest energy dominating the total energy.

 

[atyy]

IF I recall that quote comes from the PPN section? PPN is powerful, but I believe it makes some specific assumptions about the source, specifically about the rest energy dominating the total energy.

Yes. It's from the PPN section. PPN doesn't extend to truly "strong field" situations either. And even if it does, it's just another parameterization of the metric after all, so it shouldn't be that "canonical" from a pure theory point of view (wrt GR, PPN may well be canonical wrt Newton). But we will surely end up doing something non-canonical by attempting to answer this question, won't we? I mean, the only strict answer can be, as DaleSpam has emphasized, that the question doesn't make sense.

 

[PAllen]

Yes. It's from the PPN section. PPN doesn't extend to truly "strong field" situations either. And even if it does, it's just another parameterization of the metric after all, so it shouldn't be that "canonical" from a pure theory point of view (wrt GR, PPN may well be canonical wrt Newton). But we will surely end up doing something non-canonical by attempting to answer this question, won't we? I mean, the only strict answer can be, as DaleSpam has emphasized, that the question doesn't make sense.

I believe my formulation as specified in post #34 can be applied rigorously to highly relativistic motion arbitrarily close to the event horizon. And it yields a true scalar invariant as the answer given: closest approach to event horizon, speed, mass parameter of sperically symmetric static solution. I wrote down the equations for it in Ben Niehoff's metric, but the result is mathematically intractable, though perfectly well defined.

Note, the post #34 formulation is the same as #18. #34 is described in coordinates where the massive body is moving relativistically; #18 is described in coordinates where the accelerometer is moving relativistically. They are identical in physical measurement. #34 is easier to write equations for given Ben Niehoff's boosted coordinates.

 

[pervect]

I believe my formulation as specified in post #34 can be applied rigorously to highly relativistic motion arbitrarily close to the event horizon. And it yields a true scalar invariant as the answer given: closest approach to event horizon, speed, mass parameter of sperically symmetric static solution. I wrote down the equations for it in Ben Niehoff's metric, but the result is mathematically intractable, though perfectly well defined.

Note, the post #34 formulation is the same as #18. #34 is described in coordinates where the massive body is moving relativistically; #18 is described in coordinates where the accelerometer is moving relativistically. They are identical in physical measurement. #34 is easier to write equations for given Ben Niehoff's boosted coordinates.

You'll measure something, though it'll be tough to calculate. If we apply similar ideas to the curved surface of the Earth, though, we might extend lines fermi-normal from the equator towards the poles, the equator being the closest thing we can find to "flat" space. And then we might declare/decide as a result of this construction that circles of constant lattitude are "the true straight line", and calculate the relative acceleration of geodesics to our "straight" lines of constant lattitude. And we could talk about the mysterious force that tends to pull geodesics back towards the equator.

 

[pervect]

I"ve been sort of playing with a totally different idea, but so far I haven't quite been able to figure out how to make it work.

The idea is to pretend that everything in the universe has a charge, so that there aren't any electrically neutral objects. Then, we could try to define the electric field by its gradient, rather than by the field itself, as we are forced to do with gravity. And compare the resulting formulation with GR- I'm sure it will be different, because in the end E&M is a 2-form and gravity isn't.

 

[PAllen]

You'll measure something, though it'll be tough to calculate. If we apply similar ideas to the curved surface of the Earth, though, we might extend lines fermi-normal from the equator towards the poles, the equator being the closest thing we can find to "flat" space. And then we might declare/decide as a result of this construction that circles of constant lattitude are "the true straight line", and calculate the relative acceleration of geodesics to our "straight" lines of constant lattitude. And we could talk about the mysterious force that tends to pull geodesics back towards the equator.

I agree the prescription isn't unique or applicable to general spacetimes. However, given asymptotically flat spacetime, it anchors 'same course' paths to distant flat geodesics via maintaining fixed proper distance. Your sphere would be an example where the method couldn't be used - no asymptotic flatness. Though not solvable (at least by me), I can see from the resulting equations that the 4 acceleration always points in the expected direction. I also get gamma^2 factors in the equations, so it seems plausible (but not proven) that to one or two leading orders you get Bill_k's result.

Unfortunately, even in the spherically symmetric static solution, it is clear the equations for the world line cannot be directly expressed in terms of elementary functions. As a result, I don't even have 4 velocity of the world line in closed form.

 

[Ben Niehoff]

PAllen, I don't think your construction is meaningful. Basically you are describing a way to construct a coordinate system, and then asking what forces are felt by observers who maintain constant coordinates in this coordinate system.

But I don't think your coordinates have any physical meaning, and therefore I don't think the path traveled by your test mass has any sort of "naturalness" that would make it preferred over any other path for making this kind of measurement.

I really think if we want to understand the effect on the strength of gravitational attraction due to relative motion, we are pretty much limited to two options:

1. Take a test body in an unbounded orbit and compute the tidal forces felt at closest approach; compare this to the object's tangential velocity at closest approach (which can be measured relative to the timelike Killing vector of the Schwarzschild geometry), or

2. Put a test body in motion along a straight line in the asymptotic region and do linearized gravity to find the lowest-order contribution due to velocity. (This is what Bill K did.)

The reason we are limited to these options is because in GR, gravity does not exert force on a test body. A test body moving on a geodesic feels NO net force. Only tidal forces are real.

PAllen is describing a scheme by which we measure the forces on a test body not moving on a geodesic. But there is no natural, preferred way to choose a non-geodesic path, because all non-geodesic paths have local accelerations, and there is no way to distinguish "acceleration due to gravity" from "acceleration due to curvilinear motion".

(Note, there is one exception to this: in a stationary spacetime, there is a natural way to define a stationary observer. But we are asking about observers "in relative motion" with respect to the black hole center.)

 

[PAllen]

PAllen, I don't think your construction is meaningful. Basically you are describing a way to construct a coordinate system, and then asking what forces are felt by observers who maintain constant coordinates in this coordinate system.

But I don't think your coordinates have any physical meaning, and therefore I don't think the path traveled by your test mass has any sort of "naturalness" that would make it preferred over any other path for making this kind of measurement.

I really think if we want to understand the effect on the strength of gravitational attraction due to relative motion, we are pretty much limited to two options:

1. Take a test body in an unbounded orbit and compute the tidal forces felt at closest approach; compare this to the object's tangential velocity at closest approach (which can be measured relative to the timelike Killing vector of the Schwarzschild geometry), or

2. Put a test body in motion along a straight line in the asymptotic region and do linearized gravity to find the lowest-order contribution due to velocity. (This is what Bill K did.)

The reason we are limited to these options is because in GR, gravity does not exert force on a test body. A test body moving on a geodesic feels NO net force. Only tidal forces are real.

PAllen is describing a scheme by which we measure the forces on a test body not moving on a geodesic. But there is no natural, preferred way to choose a non-geodesic path, because all non-geodesic paths have local accelerations, and there is no way to distinguish "acceleration due to gravity" from "acceleration due to curvilinear motion".

(Note, there is one exception to this: in a stationary spacetime, there is a natural way to define a stationary observer. But we are asking about observers "in relative motion" with respect to the black hole center.)

These are good points, but I think what I have is a 'reasonable' proposal for is how to generalize the stationary observer to an observer maintaining a 'fixed course relative to distant stars' in the case of asymptotically flat spacetime. Objectively, I have a precise definition I can apply in asymptotically flat spacetime. We can agree to disagree on how meaningful it is. I believe it s a reasonable model of how strongly and in what direction a rocket would need to fire its thrusters to maintain a fixed course (relative to distant stars) while passing a massive body.

 

[pervect]

1. Take a test body in an unbounded orbit and compute the tidal forces felt at closest approach; compare this to the object's tangential velocity at closest approach (which can be measured relative to the timelike Killing vector of the Schwarzschild geometry), or

That's the approach I took, though it was convenient to define "closest approach" in the Schwarzschild geometry.

I'm wondering if anyone else has got numbers to compare to mine?

 

[TWest]

IS 2:1error "not much?"?

Well, I was always told that less than 10% was not much.

 

[pervect]

Well, if you want that sort of accuracy (10%), you probably need to go beyond trying to retrofit Newotonian formulae, and take the leap to curved-space time and the tensors that describe it.

 

[TWest]

I suppose that is true. I just wanted a easy less math intensive way to do it but people that study physics don't get that wish do they.