Does the Gravitational Mass of a Moving Particle Increase with Inertial Mass?

[cos]

On the basis that gravitational mass = inertial mass - does the gravitational mass of a particle increase along with its inertial mass when the particle is made to move?

Bill

 

[GreenPenInc]

On the basis that gravitational mass = inertial mass - does the gravitational mass of a particle increase along with its inertial mass when the particle is made to move?

Bill

Yes and no.

If you take the point of view, as Einstein did, that inertial mass increases with velocity, then yes, the gravitational mass increases as well. An object will exert a greater gravitational pull when moving, than when it is stationary.

Modern usage, however, treats the mass as a constant independent of reference frame. After all, the 'relativistic' mass is just the same as the relativistic energy, up to a constant (which anyone working in the field sets to 1 when manipulating equations anyway). It's widely considered simpler to use the term 'mass' to refer unambiguously to 'rest mass'.

It seems we'd have a contradiction here, though: after all, a moving object will exert a stronger pull, but won't be considered to have higher mass: thus, it seems like inertial and gravitational masses would be different! The solution, of course, is that we should shift to thinking in terms of energy, not mass. It is the energy which is the source of the gravitational field, of which the mass is only part. (The mass is the great majority of the energy at low velocities, but can be dwarfed as v approaches c.)

I had the same exact question when reading through Schwartz's E&M text. I find that switching to thinking in terms of energy helped clarify matters for me.

 

[pervect]

If you have a bunch of particles (as in a gas) moving in random directions, the gravitational mass of the ensemble of particles will be higher when the particles are moving than when they are stationary.

Another way of saying this: if you have a pressure vessel containing an ideal gas, and you heat up the gas, the mass of the system (gas+pressure vessel) will increase as a result of the faster motion of the particles in the gas.

This makes it sound like the answer to your question is "yes", but that would be misleading. It is *NOT* correct to think that each moving particle has a "gravitational field" that is given by the Newtonian formula G * relativistic mass / r^2, though it is correct to think of the ensemble of moving particles as having a higher gravitational mass and gravitational field when considered as a group.

There are technical problems with even defining the notion of the gravitational field of a single moving particle using the Newtonian definition of the gravitational field, i.e. the amount of force required to hold a particle stationary. This is because *in general* there isn't any way to define what it means to hold a particle stationary in GR in a general metric. (There are some important special cases where it *is* possible to define what it means to hold a particle stationary, the Newtonian gravitational field of a pressure vessel containing a hot gas is able to be computed because it is one of these important special cases. However, the gravitational field of a single moving mass is *not* one of the special cases :-().

While it is probably not in general possible to define the "Newtonian gravitational field" at a point, it is possible to define the tidal gravitational field at a point. Such tidal gravitational fields can actually be measured by devices such as Robert Forward's "gravity gradiometer". The tidal gravitational field of a moving particle is not spherically symmetrical, it is stronger in some directions than others. In the limit as the moving particle approaches the speed of light, the tidal gravitational field of a moving particle is concentrated almost entirely in the direction transverse to the direction of motion. This makes the "tidal gravitational field" of an ultra-relativistic particle similar to that of an impulsive gravity wave. The GR solution for the limiting case is called the Aichelburg-sexl solution. See for instance http://arxiv.org/abs/gr-qc/0110032 for the highly technical details of that solution.

For some idea of what it takes to calculate the tidal gravitational field of a moving mass at an arbitrary velocity (rather than in the limiting case for the velocity being very large, which is discussed in the above paper) see for example the following posts:

https://www.physicsforums.com/showpost.php?p=690472&postcount=10
https://www.physicsforums.com/showpost.php?p=1347429&postcount=10

which are also highly technical