Proper mass and space-time wrap 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?

• sorry for some reason the text didn't turn out organized as i wrote it. – Raeed mndow Jun 18 '17 at 20:12
• You gotta put two line breaks between paragraphs. – Javier Jun 18 '17 at 20:37

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.