Gravitational Lensing of Fast Receding Galaxies: Does Speed of Light Matter?

[Einstein's Cat]

I am aware that the greater a body's velocity the greater its relativistic mass. As a result, I assume that the faster a galaxy is receeding from us, the greater its gravitational lensing affect is. My question is this: does the extent of a galaxies gravitational lensing continue to increase at a constant rate as the galaxy's recessional velocity exceeds the speed of light or not? Thank you for your help and please correct any stupid assumptions!

 

[Orodruin]

As a result, I assume that the faster a galaxy is receeding from us, the greater its gravitational lensing affect is.

You then assume wrong. Relativistic mass is an archaic concept which is generally no longer referred to by professional physicists. It has nothing to do with how gravitation works.

 

[Einstein's Cat]

You then assume wrong. Relativistic mass is an archaic concept which is generally no longer referred to by professional physicists. It has nothing to do with how gravitation works.

Why is the concept no longer referred to?

 

[Ibix]

Why is the concept no longer referred to?

Because it leads to conversations like this. So called "relativistic mass" has nothing to do with gravitation. The source term for gravity in general relativity is the stress-energy tensor, which does not include a term that looks like relativistic mass.

Typing this on my phone. Predictive text successfully typed the whole of that last sentence from "stress-energy" onwards. I may have answered this question before...

Edit: The stress-energy tensor does include a term for mass (or "rest mass", if you prefer). That's why Newtonian gravity has mass as the source term. There are also terms in the stress-energy tensor that are affected by the motion - just not in the straightforward way you may be imagining. For more info, have a search on this site for "what is relativistic mass and why isn't it used".

 

[pervect]

I am aware that the greater a body's velocity the greater its relativistic mass. As a result, I assume that the faster a galaxy is receeding from us, the greater its gravitational lensing affect is. My question is this: does the extent of a galaxies gravitational lensing continue to increase at a constant rate as the galaxy's recessional velocity exceeds the speed of light or not? Thank you for your help and please correct any stupid assumptions!

Unfortunately, relativistic mass does not directly determine how much a body gravitates, specifically plugging "relativistic mass" into Newton's gravity law doesn't give the correct answers.

There is a paper that addresses the effect of a moving body on a motionless dust field that gives the correct relativistic answer to how much velocity is imparted to a stationary dust field by a relativistic flyby, and compares it to the Newtonian answer. See Olson, D.W.; Guarino, R. C. (1985). "Measuring the active gravitational mass of a moving object". When compared in this particular manner, the relativistic answer is that you get more of an induced velocity due to a relativistic flyby than you would by (incorrectly) using Newton's laws and plugging in the "relativisitc mass" in place of the Newtonian M.

But I don't know of anyone who has derived a similar result for gravitational lensing. I suspect that the result will be similar to Olson's for dust, but I can't guarantee it . You may have some difficulty tracking the paper down, though. You can find the abstract of the paper at http://adsabs.harvard.edu/abs/1985AmJPh..53..661O, I'll attempt to quote the abstract (but it's better to read the full paper if you can get it).

If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic increase in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that Mrel=γ(1+β2)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.

The abstract is a bit over-general, it's important to realize that the "active gravitational mass" is a term not (AFAIK) in general use, specific to this paper, that computes what one might term the "average" field due to a relativistic flyby. This is something that is explained well in the paper but isn't clear at all from the abstract. The more commonly used notions of mass in GR are such masses as the Komar mass, the Bondi mass, and the ADM mass, which behave differently than the sort of "mass" the author describes in this paper.

Something else that is interesting to note and may not be mentioned in this paper, though it's mentioned in other papers. If you look at the peak field (and not the average field) of an ultra-relativistic flyby, you get the Aichelberg sexl solution, which is an impulsive plane-polarized gravity wave. In the ultra-relativistic limit, the velocity of a test particle changes by an impulse function, not in a continuous manner.

 

[Einstein's Cat]

Unfortunately, relativistic mass does not directly determine how much a body gravitates, specifically plugging "relativistic mass" into Newton's gravity law doesn't give the correct answers.

There is a paper that addresses the effect of a moving body on a motionless dust field that gives the correct relativistic answer to how much velocity is imparted to a stationary dust field by a relativistic flyby, and compares it to the Newtonian answer. See Olson, D.W.; Guarino, R. C. (1985). "Measuring the active gravitational mass of a moving object". When compared in this particular manner, the relativistic answer is that you get more of an induced velocity due to a relativistic flyby than you would by (incorrectly) using Newton's laws and plugging in the "relativisitc mass" in place of the Newtonian M.

But I don't know of anyone who has derived a similar result for gravitational lensing. I suspect that the result will be similar to Olson's for dust, but I can't guarantee it . You may have some difficulty tracking the paper down, though. You can find the abstract of the paper at http://adsabs.harvard.edu/abs/1985AmJPh..53..661O, I'll attempt to quote the abstract (but it's better to read the full paper if you can get it).
The abstract is a bit over-general, it's important to realize that the "active gravitational mass" is a term not (AFAIK) in general use, specific to this paper, that computes what one might term the "average" field due to a relativistic flyby. This is something that is explained well in the paper but isn't clear at all from the abstract. The more commonly used notions of mass in GR are such masses as the Komar mass, the Bondi mass, and the ADM mass, which behave differently than the sort of "mass" the author describes in this paper.

Something else that is interesting to note and may not be mentioned in this paper, though it's mentioned in other papers. If you look at the peak field (and not the average field) of an ultra-relativistic flyby, you get the Aichelberg sexl solution, which is an impulsive plane-polarized gravity wave. In the ultra-relativistic limit, the velocity of a test particle changes by an impulse function, not in a continuous manner.

Thank you so much for your help which has proven very useful!