Proper mass and space-time wrap question

Question

I still haven't tackled GR equations because I still don't have the mathematical tools, but that didn't stop me from looking into it. I have a question and was hoping someone would clear things up for me. Tell me if I made a mistake along the way:

• As far as I know the metric tensor is invariant meaning the curvature of space looks the same for all frames.

• I think that means that "relativistic mass" doesn't determine the curvature but the rest mass does.

• The rest mass is the total internal energy in the object divided by $c^2$. right? My mass is basically the energy of the particles in my body.

• So I concluded that even if I speed up and move at high speed I won't curve space any further than I did when I'm at rest. Otherwise if I get real close to light speed I would eventually would turn into a black hole.

• Photons do not have rest mass, but you can put a photon in a mirror box (0 mass) and see that the mass of the box would turn out to be $hf/c^2$.

Here is my question: I've read everywhere that photons do in fact bend space and contribute to the stress tensor, very very very little tho, so how can that be possible if their rest mass is zero? Are massless objects treated differently because their rest frame of reference is kinda weird?

Answer 1

Nice question. I'll try to answer, but as always the sad truth (or happy truth, that's up to you) is that to fully understand these things you just gotta learn the math.

This question is a bit tough to answer because the metric is not invariant, it's covariant. What this means is that in a given coordinate system the metric is represented as a matrix, and the components of this matrix depend on your coordinates, even if the metric tensor as a mathematical object doesn't. This is much like how you can think of a vector as something that exists independently of the frame, but if you want its components, they do depend on the frame.

The reason this makes the question hard to answer is that "curvature" is not a coordinate independent number. There are some scalar (i.e. invariant) quantities you can form like the Ricci scalar $R$ or the Kretchmann scalar $R_{abcd} R^{abcd}$, but these don't tell the whole story. In other words, there isn't an unambiguous "curvature" which you can use to say that this place has high curvature and that place has low curvature.

Now to your question: what determines curvature is the energy momentum tensor. This includes the energy density and also the momentum density/energy flow and the pressure. For a point particle the energy density is the relativistic mass, so your speed does matter if you want to calculate the metric tensor. Similarly, it doesn't matter that light is massless, because we care about its energy and momentum, not its mass.

The conclusion here is that the curvature tensor is, in a sense, dependent on the frame. However, some things are not dependent of the frame. For example, the formation of a black hole. If you're not a black hole when standing still then you're not going to become one when moving, because of relativity. Even though your energy increases, everything works out in such a way that the formation of a black hole depends only on invariant quantities.