I Velocity, Gravitational Pull & Creator's Movement: Relation?

[kos]

How does movement of the creator of gravity field( mass or energy density) affect the magnetude of exerted gravitational field ! Is there any relation at all ?

 

[PeterDonis]

Movement is relative, so I assume you are asking how the field detected by an object that is moving relative to the source differs from the field detected by an object at rest relative to the source; is that correct?

 

[kos]

Yes

 

[Jonathan Scott]

How does movement of the creator of gravity field( mass or energy density) affect the magnetude of exerted gravitational field ! Is there any relation at all ?

Unless you're dealing with an extreme case (neutron stars or worse), or need extreme accuracy, you can work this out roughly using Special Relativity, as the resulting motion has to be consistent when seen from different frames of reference.

Work out what it looks like from the frame where the gravitational source is at rest and the test object is moving, then Lorentz transform the resulting acceleration to the frame of the test object (noting the extra time dilation factor for transforming acceleration rather than just velocity).

For motion sideways (tangentially) relative to the field, you can use the fact that the radial acceleration of a test mass in General Relativity is approximately $(1 + v^2/c^2)$ times the Newtonian acceleration.

Sorry I can't be bothered to work out the details, but if you have a go I'm sure someone will help if necessary!

 

[pervect]

I'd recommend (as I have in the past when this question has come up) the following paper. http://scitation.aip.org/content/aapt/journal/ajp/53/7/10.1119/1.14280 "Measuring the active gravitational mass of a moving object".

A very brief summary. In GR, there are some hard-to describe and non-intuitive effects due to various sorts of curvature. These issues can be sidestepped by looking at the total velocity imparted to a stationary mass, initially at rest in some inertial frame, by a relativistic flyby. Thus test mass is at rest before the flyby in some inertial frame, the flyby occurs during which "stuff happens" and the frame isn't inertial, then after the flyby the frame is inertial again and the perturbing effect on the velocity of the test object is measured.

The net result is that the imparted velocites are very similar when alanlyzed with GR and Newton, with an "effective mass" parameter that is a function of the velocity and mass of the gravitating object.

To quote the result from the abstract of the paper:

then we find, with this definition, that M_effective = γ(1+β^2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not γM but is approximately 2γM.

It's worth reading the whole paper and not just the abstract, if you can find it.

Another pitfall to avoid is thinking that the gravitational field due to a relativistic flyby is spherically symmetric. It is not. In the ultra-relativistic limit, it becomes an impulsive function, i.e. the test mass suddenly changes direction. The GR solution for this case is illuminating but much more technical than the Olson paper, as the effects of curvature have to be taken into account. The GR solution describing this case is called the Aichelbrg Sexl ultraboost. There is a rather technical discussion in wiki under https://en.wikipedia.org/wiki/Aichelburg–Sexl_ultraboost

 

[Jorrie]

I'd recommend (as I have in the past when this question [effect of velocity on gravity] has come up) the following paper. http://scitation.aip.org/content/aapt/journal/ajp/53/7/10.1119/1.14280 "Measuring the active gravitational mass of a moving object".

(squared brackets my insert for context)
I found the analysis that you have done in an old thread (https://www.physicsforums.com/threa...in-cartesian-coordinates.126996/#post-1046874) quite illuminating on the subject, where you have derived the Schwarzschild coordinate accelerations of a particle orbiting in a static gravitational field.
\frac{d^2 r}{d t^2} = \frac {3 m{{\it vr}}^{2}}{ \left( r-2\,m \right) r} + \left( r-2\,m \right) \left( {{\it vphi}}^{2}-{\frac {m}{{r}^{3}}} \right)
\frac{d^2 phi}{d t^2} = -\frac {2 {\it vr}\,{\it vphi}\, \left( r -3\,m \right) }{ \left( r-2\,m \right) r}
with vr = dr/dt and vphi = dphi/dt and c=G=1.

If I am not mistaken, this can be generalized to any reasonable* radial and transverse speed combination relative to the Schwarzschild mass m, with v_r as you had it and v_t = r vphi.
For the radial coordinate acceleration
\frac{d^2 r}{d t^2} = \frac {3 m{{\it v_r}}^{2}}{ \left( r-2\,m \right) r} + \left( r-2\,m \right) \left( {\it r^2 v_t^2}-{\frac {m}{{r}^{3}}} \right)
This can be regrouped to give
\frac{d^2 r}{d t^2} = -\frac {m}{r^2} \left(1-2m/r- \frac{3v_r^2}{1-2m/r} + 2v_t^2 \right)

This is essentially the modification to the Newtonian acceleration (-m/r²) due to curvature and velocity in Schwarzschild coordinates.

Note* I think there is a limit to the v_r and v_t combination, because the proper time
d\tau = (1-2m/r-v^2)^{0.5}dt, must remain real, where v is the locally measured velocity (not Schwarzschild).

 

[pervect]

Jorrie: You might get some intuition from the way you're doing things, but it's a bit risky to interpret $d^2 r / dt^2$ as being due to a "force". If you work out the problem in, say, isotropic coordinates, you will probably get an answer that looks more complicated, for instance, and you might get a different interpretation. Having a different interpretation for every coordinate system you might want to use becomes problematic.

In this particular case, I'd say the isotropic coordinates were more physically meaningful coordinate choice in general - but I'd expect the math to be a lot messier :(.

The issue is abstract, but fundamental. Forces are supposed to transform as rank 1 tensors. But GR doesn't have any rank 1 tensors that match up to the idea of "forces". The closest thing are probably Christoffel symbols, which aren't tensors - and have 3 indices, not just 1. So they don't transfom in the same manner as forces do when you change coordinates.

The significance of this is that when you make a statement about forces, the tensor nature of forces makes the statement true in all coordinate systems. When you make a statement about Christoffel symbols, the statement is true only in the coordinate system in which you derived it, it will in general have a different appearance if you make a different coordinate choice.

It may be tempting to solve this dilemma by insisting that everyone always use a specific coordinate system. I see a lot of PF posters, for instance, who regard the Schwarzschild coordinates as "the coordinate system of choice", and won't use any other. The difficultes arise when someone else uses a different coordinate system - their results are prefectly valid, but communication doesn't happen. One specific issue with insisting on using Schwarzschild coordinates for everything is that one tends to run into problems at the event horizon, because the Schwarzschild coodinates are really ill-behaved there. This often leads to long confused threads, where the ill-behavior of Schwarzschld coordinates is confused with or interpreted as the ill-behavior of the physics, when the facts of the matter (as most textbooks and a lot of PF posts will indicate) that it's not the physics that's ill-behaved.

The issue could be and is in the literature resolved simply by choosing different coordinates - but this procedure isn't recognized by posters who have limited themselves to considering only one choice.

 

[Jorrie]

Jorrie: You might get some intuition from the way you're doing things, but it's a bit risky to interpret d2r/dt2" style="font-size: 112%; position: relative;" tabindex="0" class="mjx-chtml MathJax_CHTML" id="MathJax-Element-7-Frame">d2r/dt2 as being due to a "force".

I understand and agree with what you said, but as long as we avoid the term 'force' and also emphasize that it is a specific coordinate acceleration and only valid for r > 2m (which I did not do), I see no harm done. And the equation is so beautifully simple...