# Relativistic Mass and Gravity

Does relativistic mass affect space time in the same way as rest mass?

To my understanding, (as I am not an actual physicist, but simply a citizen scientist) relativistic mass is really the measure of an object's energy. It is not the same as rest mass, which is the definition of mass that a layperson would be familiar with (how much matter an object is composed of). However, does a change in relativistic mass amount to the same magnitude of gravitational variation as an equivalent change in rest mass?

In General relativity , it is the stress energy tensor that defines space time curvature. Thus mass is a secondary definition.

The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

Relativistic mass is not a good concept in defining a particles energy and momentum which will enter the stress energy equations, although it started from the T00 component of the stress energy tensor, by defining an energy density.

Excess velocity will give excess energy and momentum, the same as excess Newtonian mass, but one should keep a distinction in the concepts to avoid confusion.

You are right that relativistic mass is just another name for total energy (mass-energy + kinetic-energy). This is why most physicists no longer user the term relativistic mass as it is redundant and causes confusion.

To answer your question about gravity, let's imagine a star with a planet in orbit around it. There are two physicists observing this solar system in two different spaceships: one is at rest with respect to the star near a point in the planet's orbit, the other is traveling at high speed past the system. They both have clocks to measure the period of the planet's orbit. Due to time dilation, the stationary physicist will measure a shorter period than the moving physicist since, in the high-speed physicist's frame, the star and planet are moving, and so move through time more slowly.

But, from the high-speed physicist's perspective, the star and planet are moving at high speed. If relativistic mass affected the gravitational force, then we should expect the period of the planet to be shorter, since the mass of the star would be greater: $$T = \sqrt{\frac{4\pi r^3}{GM}}$$ where $T$ is the orbital period, $r$ is the radius of the circular orbit, $G$ is the gravitational constant, and $M$ is the mass of the star.

Increase the speed of the frame to arbitrarily close to the speed of light, and the planet's orbit gets shorter and shorter, even as time dilation should cause the planet to move slower and slower. This contradiction tells us that relativistic mass can't affect gravity.

For a second example, imagine the planet orbits a black hole. From a viewpoint at rest with respect to the black hole, the planet is fine. From a high speed perspective, the black hole's relativistic mass increases, increasing its size. At a high enough speed, the black hole is large enough that it swallows the planet. Meanwhile, the planet is still fine from the stationary perspective. Again, relativistic mass cannot affect gravity.

• I have sort of a appendix question to this. Does relativistic mass (say the energy of a photon) affect spacetime? As a thought experiment imagine a perfect reflective sphere with a single point of light at the center, what would happen as the cavity is filled with photons? Would any time dilation occour inside? – EJTH Sep 23 '18 at 15:22
• @EJTH To find out the affect of a system on spacetime, pick the reference frame where momentum is zero. For a single photon, there is no such reference frame, so the effect on spacetime is zero. Now, two photons heading directly towards each other do have a reference frame with zero momentum. So, when they pass each other, they will have a gravitational field. If they collide, they may form a particle-anti-particle pair, which will definitely have a gravitational field. Your example of a sphere containing photons is the same: there is a zero-momentum frame, so there is a gravitational field. – Mark H Sep 26 '18 at 1:56
• @EJTH As for time dilation, I'm not sure what you're referring to. What time would be dilated? – Mark H Sep 26 '18 at 1:56