# Gravitational Field of a Photon compared to that of Massive Matter

[I'm aware relativistic mass is an outdated term, but I'm not sure what term to use in place of it]

How is it that [as I've heard, perhaps incorrectly] photons can contribute to the stress-energy tensor and thus interact gravitationally purely through their relativistic energy $E = pc$ and their associated relativistic mass from that, whereas if you have a massive particle of rest mass $m_0$ and relativistic mass $m_1$ only the rest mass affects the gravitational field? What distinguishes the relativistic masses of these particles to the point where one has an effect and the other doesn't?

• You seem to be suggesting that a moving massive particle has the same stress-energy tensor (or "effect on the gravitational field") as a massive particle at rest. Where have you heard this? – Prahar Sep 17 '17 at 7:01
• How is it that [as I've heard, perhaps incorrectly] photons can contribute to the stress-energy tensor and thus interact gravitationally purely through their relativistic energy E=pc and their associated relativistic mass You're right, this is incorrect. They also contribute through their pressure. So basically this question is of the form "Why is it true that A and B," where both A (as pointed out by Prahar) and B (this) are false. – Ben Crowell Sep 17 '17 at 15:28
• Thanks, sorry for asking a crappy question, I'll take information from the person who told me both of the conditions with a pinch of salt next time. – user95137 Sep 17 '17 at 20:25

The curvature of spacetime depends on the metric tensor. A tensor is an object that does not depend on the coordinate system, although its components are different in different coordinate systems. This is similar to a vector: if you change the coordinate system, the components of a vector would change, but the vector itself would not (except of course for the radius vector).

For a non-accelerating object that is relativistic in your frame, the object is stationary in its proper frame. The metric tensor in this frame is defined by the invariant mass (a.k.a. rest mass). Now, in your frame, the components of this tensor will be different, but the tensor itself, as an object defining the spacetime curvature, would be the same.

This is different for light, because it is impossible to define an inertial frame, in which light is stationary. It gets even more interesting from here. The photon does not generate a static gravitational field. The gravitational field of the photon is a gravitational wave perpendicular to the direction of the photon and moving with the speed of light.

The fact that both the photon and its gravitational wave move with the speed of light works out to a remarkable result. From the viewpoint of the observer affected by this gravitational wave, the wave is not associated with the photon, but with the spacetime events of the photon emission and absorption, where the former is seen as an attraction while the latter as a repulsion:

The gravitational field of photons

The gravitational field of a laser pulse

The result that the gravitational wave of the photon is associated with the emitter is remarkable, because photons do not experience time and therefore cannot change in flight (e.g. emit gravitons). This is similar to the fact that neutrino oscillations are the evidence that neutrinos do not move with the speed of light. However, a full understanding of the gravity of light requires developing a theory of quantum gravity.

You can often hear an argument that light trapped in a massless box with ideal internal reflection has invariant (rest) mass and therefore creates the same gravitational field as a massive object of the same mass. This is incorrect. The gravitational field would not be the same.

• it reminds me of lasers, when trying to describe interference the whole lasing system has to be taken into account when looking at energy balances youtube.com/watch?v=RRi4dv9KgCg . photons are strange beasts – anna v Sep 17 '17 at 14:50
• "You can often hear an argument that light trapped in a massless box with ideal internal reflection has invariant (rest) mass and therefore creates the same gravitational field as a massive object of the same mass." Take the pi0. It has an invariant mass, and the consequent gravitational field.It decays to two photons with the same invariant mass.Maybe we have to wait for a definitive quantization of gravity to say whether the invariant mass of the two photons interacts the same way gravitationally as the parent pion ( which seems the reasonable assumption as the decay has no gravitons). – anna v Sep 17 '17 at 14:57
• @anna v: Agreed, although quantum gravity may not look the way it's expected. I don't see how a photon can emit gravitons in the middle of the flight, because the flight has no duration for the photon. The same way as oscillations are impossible for massless neutrinos. So the idea in the links that the gravitational wave is actually associated with the events of the photon emission and absorption is interesting. It is similar to the fact that the Earth is attracted to the actual location of the Sun, but not to where the Sun was 8 minutes ago. – safesphere Sep 17 '17 at 22:13