Protons have mass and as a result of einstein's field equation dictate that the spacetime is no longer flat. But yet I find in most Quantum Field Theory books the Minkowski flat spacetime metric is used.

Doesn't this imply that the space is flat even with the proton being present? Or are we making an approximation? But if it's an approximation, looking at success of QFT it looks like that's a really good approximation to make.

I am unable to picture an electron in spacetime without it bending it. How does one think about this ?

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    $\begingroup$ It seems like you are asking about a formulation of quantum gravity, which has not been fully established yet. $\endgroup$ – BioPhysicist Dec 14 '18 at 0:51
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    $\begingroup$ I guess a better question would be why does QFT which is in Minkowski space work so well ? $\endgroup$ – 256ABC Dec 14 '18 at 1:15

It's an approximation, but a really good one. The reason it's really good can be seen by just considering how absurdly weak gravity is. For an electron gravitational effects are about $10^{43}$ times weaker than electrostatic ones, for a proton it's about $10^{37}$ (protons are more massive than electrons!)

So just to detect the gravitational interactions of, say, electrons, you would need to be able to design an experiment which could somehow notice this $\sim 10^{-43}$ effect.

It turns out that it's really easy to do such experiments: you get very large numbers of electrons, protons and neutrons and construct large, electrostatically neutral, objects out of them like planets and stars and people. And suddenly all the electromagnetic stuff cancels out and you can measure gravity.

But doing this on the scale of individual electrons or protons is almost impossibly (I hesitate to say actually impossibly) hard, because gravity is so weak.

Of course the fact that it is so hard is one of the reasons why quantum gravity is so hard: if we could easily do particle experiments where gravitational effects were easy to detect, then it would be easy to test theories of quantum gravity.

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