Gravitational Mass & Velocity: Is There a Correlation?

[Jan Nebec]

Hello!
The equation for relativistic mass by special relativity tells us the relativistic mass for object in motion...but since inertia mass has same value as gravitational, does this formula also apply for gravitational mass?
Thank you!

 

[Nugatory]

It does not, and that is just one of many reasons why over the past few decades physicists are abandoning the entire notion of relativistic mass as a poor way of thinking about what's going on: https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

 

[pervect]

Hello!
The equation for relativistic mass by special relativity tells us the relativistic mass for object in motion...but since inertia mass has same value as gravitational, does this formula also apply for gravitational mass?
Thank you!

The short answer is no. One difficulty here is that to talk about gravity, one usually needs general relativity. And General relaltivity (GR) has several different notions of mass - the concept of mass in GR is a rather complex and subtle subject.

There is at least one approach that gives some useful answers to this question that avoids the need to talk about the curvature of space-time, a hallmark of GR. This approach considers a cloud of test particles in a perfectly flat space-time of special relativity. The cloud is perturbed by a relativistic flyby of a massive object. Calculating what happens during the flyby requires the techniques of GR. However, after the massive object has completed it's flyby, the space-time is once again flat. And one can usefully ask - how are the velocities of the test particles in the cloud perturbed by the flyby, avoiding all the difficulties of dealing with a curved space-time.

Olson & Guarinio <<link>> have a paper in which they discusses this situation in some detail.

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+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.

Potentially of some interest is an old paper by Misner, https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1045 that also talks about "active gravitational mass". It appears to come to a similar conclusion, though I don't have the complete paper to read (just the abstract). Misner's paper talks about why the "active gravitational mass" of a system containing a pair of anti-particles doesn't change when the particles annihilate. The explanation that Misner gives is that it's the tension in the wall of the container containing the annihilation products that explains this result.

Misner's paper points out something important that we often mention on physics forum. In GR it is not just "mass" that causes gravity. Rather it is the stress-energy tensor. The stress energy tensor includes effects such as momentum, pressure, and tension (which is just pressure with the opposite sign). So if one is trying to understand the GR results, one will not get a complete and correct understanding of it if by assumes that "mass causes gravity" as it does in Newtonian physics. Unfortunately, the details of understanding exactly what the stress-energy tensor is can be intimidating. At the beginner level, about all I can usefully say is that energy, momentum, and pressure all contribute to the stress-energy tensor, and that we replace the idea that "mass causes gravity" from Newtonian theory with the idea that "the stress-energy tensor" causes gravity in GR. Miser's paper, for instance, just doesn't make sense unless one understands that in GR, tension in the walls of a container can cause gravity, something that isn't true at all in Newtonian physics.

 

[Jan Nebec]

The short answer is no. One difficulty here is that to talk about gravity, one usually needs general relativity. And General relaltivity (GR) has several different notions of mass - the concept of mass in GR is a rather complex and subtle subject.

There is at least one approach that gives some useful answers to this question that avoids the need to talk about the curvature of space-time, a hallmark of GR. This approach considers a cloud of test particles in a perfectly flat space-time of special relativity. The cloud is perturbed by a relativistic flyby of a massive object. Calculating what happens during the flyby requires the techniques of GR. However, after the massive object has completed it's flyby, the space-time is once again flat. And one can usefully ask - how are the velocities of the test particles in the cloud perturbed by the flyby, avoiding all the difficulties of dealing with a curved space-time.

Olson & Guarinio <<link>> have a paper in which they discusses this situation in some detail.
Potentially of some interest is an old paper by Misner, https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1045 that also talks about "active gravitational mass". It appears to come to a similar conclusion, though I don't have the complete paper to read (just the abstract). Misner's paper talks about why the "active gravitational mass" of a system containing a pair of anti-particles doesn't change when the particles annihilate. The explanation that Misner gives is that it's the tension in the wall of the container containing the annihilation products that explains this result.

Misner's paper points out something important that we often mention on physics forum. In GR it is not just "mass" that causes gravity. Rather it is the stress-energy tensor. The stress energy tensor includes effects such as momentum, pressure, and tension (which is just pressure with the opposite sign). So if one is trying to understand the GR results, one will not get a complete and correct understanding of it if by assumes that "mass causes gravity" as it does in Newtonian physics. Unfortunately, the details of understanding exactly what the stress-energy tensor is can be intimidating. At the beginner level, about all I can usefully say is that energy, momentum, and pressure all contribute to the stress-energy tensor, and that we replace the idea that "mass causes gravity" from Newtonian theory with the idea that "the stress-energy tensor" causes gravity in GR. Miser's paper, for instance, just doesn't make sense unless one understands that in GR, tension in the walls of a container can cause gravity, something that isn't true at all in Newtonian physics.

Huh...well thank you for this long and comprehensive answer! I didn't know about "stress-energy tensor". I guess I'll have to leave my question for some time later when I finish my studies.

 

[Jonathan Scott]

I'd phrase the answer differently. In the trivial case, the overall gravitational source strength of a stable approximately spherical configuration of matter and energy (including internal energy such as rotation or particles bouncing around in a box) as seen from a distance is still essentially proportional to its total energy in Special Relativity, including internal kinetic energy, minus the Newtonian potential energy. To that extent, GR leads on from Newtonian theory. However, in General Relativity the gravitational laws of motion are much more complex than in Newtonian theory, so the effect depends on velocity and acceleration as well as position, and for more detailed calculations involving less symmetrical situations it is necessary to consider the full stress-energy tensor.

 

[pervect]

I'd phrase the answer differently. In the trivial case, the overall gravitational source strength of a stable approximately spherical configuration of matter and energy (including internal energy such as rotation or particles bouncing around in a box) as seen from a distance is still essentially proportional to its total energy in Special Relativity, including internal kinetic energy, minus the Newtonian potential energy. To that extent, GR leads on from Newtonian theory. However, in General Relativity the gravitational laws of motion are much more complex than in Newtonian theory, so the effect depends on velocity and acceleration as well as position, and for more detailed calculations involving less symmetrical situations it is necessary to consider the full stress-energy tensor.

I don't view things this way. If we stick to special relativity, I would argue that the idea of the mass of a point particle comes from knowing the total energy and total momentum of said particle. Then, when we choose a frame of reference such that the total momentum is zero, the particle is characterized by it's energy in that frame, which we call the rest energy. When we scale the rest energy by the appropriate factor of c^2, the rest energy becomes the mass, which can also be regarded as an invariant of the energy-momentum tensor of a point particle.

The analysis needs a bit of work to extend it from point particles to distributed systems. This eventually involves replacing the energy-momentum tensor that we used for a point particle with the stress-energy tensor we use for a distributed system. But I won't go into the details of this extension. So far, this is all special relativity, so when we move from point particles to distributed the systems, when we take a tensor approach, we wind up needing the stress-energy tensor, the same tensor we use in GR. I'm not aware of any really good way to analyze the problem of a distributed system without tensors.

It is worth looking quickly at the difficulties that arise when one has a distributed system (such as a rod), rather than a point particle, in special relativity. One blithely talks about "the energy of the rod", or "the momentum of the rod" as if it were a point particle, but the rod is not a point particle. Of course, the answer to this seems obvious. When one talks about "the energy of the rod", one means the energy of the rod "now", at some specific instant of time.

However, in special relativity, the notion of "now" is frame dependent. So when one chooses a different frame of reference, the notion of "now" that one uses to do the summation also changes.

Tensor methods allow us to analyze the problem in ways that don't depend on which notion of "now" we use. Non tensor methods do not guarantee this, and the very notion of "the energy" of a system can turn out to be unexpectedly frame dependent if care is not taken.

These difficulties are not just abstract theory, they show up in problems as simple as the mass of an accelerating rod in special relativity. However, I don't think there is any good way to talk about how to resolve the difficulties without a familiarity with tensors. And as I remarked earlier, given a famliarity with tensors, it becomes clear that the invariant mass, the energy-momentum tensor, and the stress-energy tensor are the tensor quantities of interest.

Further wrinkles appear when we want to move onto general relativity, but I think the post is long enough already so I won't go into them.

 

[vanhees71]

Well, I cannot resist to point out that this is a perfect example for the harm done when students are taught the outdated concept of "relativistic mass" in introductory SR (see the other thread https://www.physicsforums.com/threads/usefulness-of-relativistic-mass.943464/ )!

 

[Jonathan Scott]

The main problem with "relativistic mass" is simply the outdated terminology. In SR and GR, what used to be called "relativistic mass" is the total energy expressed in mass units. In simple cases, that does match up with the gravitational source strength.

In some cases (for example when using isotropic coordinate systems where the coordinate speed of light varies from the standard value) it is quite useful to have separate terms for the total coordinate energy of an object at rest (which varies with time dilated clock rate) and the equivalent in coordinate mass units (which varies by the inverse cube of the clock rate).

 

[vanhees71]

In SR I can agree with your analysis, but in GR? There the sources are not mass (of whatever kind) nor energy only but the full energy-momentum-stress tensor of the matter fields (including em. radiation). The 2nd paragraph is completely enigmatic to me :-(.

 

[Jonathan Scott]

In SR I can agree with your analysis, but in GR? There the sources are not mass (of whatever kind) nor energy only but the full energy-momentum-stress tensor of the matter fields (including em. radiation). The 2nd paragraph is completely enigmatic to me :-(.

According to standard GR in a weak approximation, the effective source strength of a stable central body is given by the total effective rest energy of the body (including internal kinetic energy and other forms of internal energy), reduced by the time dilation factor due to the gravitational potential caused by interactions between the parts making up the body (which reduces the energy by twice the Newtonian potential energy, because the potential energy between any two parts is counted for both parts) and increased by the integral of the pressure over the body (which also adds up to the Newtonian potential energy), so overall the source strength is the SR-style total energy (including internal kinetic energy) reduced by the Newtonian potential energy.

In the weak approximation in isotropic coordinates getting closer to a central body, local time runs slow compared with coordinate space and local space shrinks in size compared with coordinate space by the same fraction, so the coordinate speed of light is affected by both of those and varies by the square of the time dilation factor (or by approximately twice as much when described as a fractional change). The coordinate energy of an object depends directly on the relative time dilation, but for the coordinate mass one must divide by the coordinate speed of light squared, so the coordinate mass must vary as the time dilation factor to the power of -3. This may not be very useful in GR notation, but can be useful when describing the relationship between Newtonian and GR concepts. (In non-isotropic coordinates, the coordinate speed of light varies with direction, so in that case coordinate mass, given by total energy divided by the square of the coordinate speed of light, also varies with direction and is not a useful concept).

[Edited to clarify "fractional change" phrase]

 

[PeterDonis]

According to standard GR in a weak approximation

Just to clarify, the statement you are making here requires, not just the weak field approximation, but also that the system be stationary (and I think also asymptotically flat). Which still leaves a large class of systems that can be usefully studied in this approximation, but it's good to recognize what exactly is required for it.

 

[Jonathan Scott]

Just to clarify, the statement you are making here requires, not just the weak field approximation, but also that the system be stationary (and I think also asymptotically flat). Which still leaves a large class of systems that can be usefully studied in this approximation, but it's good to recognize what exactly is required for it.

Agreed.
I had hoped that "stable central body" would indicate this informally, but it's good to clarify. I chose "stable" rather than "static" because the body can have internal motion but can't for example be in the process of exploding or collapsing.

 

[PeterDonis]

I chose "stable" rather than "static" because the body can have internal motion but can't for example be in the process of exploding or collapsing.

Yes, that's why I said "stationary", that allows for, e.g., the body to be rotating, but not exploding or collapsing.

 

[vanhees71]

Just to clarify, the statement you are making here requires, not just the weak field approximation, but also that the system be stationary (and I think also asymptotically flat). Which still leaves a large class of systems that can be usefully studied in this approximation, but it's good to recognize what exactly is required for it.

It's also not clear, what's meant by "coordinate time". I'd discuss the Schwarzschild metric, specify the coordinates and then discuss what's going on. The weak-field limit can of course also be taken using the exact solution. I also don't understand what this has to do with the original question about mass. In the Schwarzschild solution a parameter $M=r_{\text{S}}/2$ occurs, which is the total effective mass of the spherical symmetric mass distribution, viewed from empty space, where the (outer) Schwarzschild solution applies. The use of other notions of mass are at least not easy to understand, if not physically irrelevant.

 

[Jonathan Scott]

The quantity I described as the "effective source strength" is indeed the mass parameter in the Schwarzschild solution. As an overall value, it is the effective invariant rest mass of the object as a whole. However, as in Special Relativity, it is the total energy (or the "relativistic mass") of each component part rather than its rest mass which contributes to the overall invariant mass, modified by the gravitational potential and pressure in the way which I previously described.

 

[sweet springs]

...but since inertia mass has same value as gravitational, does this formula also apply for gravitational mass?

As commented above relativistic mass is a wrong idea. Pythagoras relation of energy, momentum, and mass constant was introduced instead.
However, in a special case when total momentum is zero, your suspect might be supported.
Say you heat an iron ball, molecules vibrate highly so molecules got higher kinetic energy. Molecules vibrate random and the ball has zero momentum in total. The iron ball hold more energy and thus more inertia and more source of gravity influencing outer bodies.

Not mass but energy (precisely the tensor mentioned in above posts) matters.