Exploring the Effects of Relative Mass and Relativity on Gravitational Pull

[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.