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We know that special relativity is just a special case of general relativity and we can consider the space time to be flat in constant velocity motions like in special relativity.

If that's the case then how can general relativity managers to explain the increased mass (relativistic mass) due to relativistic effects at near light speed motion?

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In general relativity, the matter source is the stress-energy tensor $T_{ab}$, a quantity that is lorentz-covariant in special relativity, so under a change of reference frame, $T$ changes. It is set explicitly equal to another covariant term made up of spacial curvature, $R_{ab} - \frac{1}{2}R g_{ab}$, so any shifts between energy and momentum will be reflected there.

*I'd also add that most relativists believe that the notion of relativistic mass is more confusing than it is worth, and it's better to think of kinetic energy, momentum, and invariant/rest masses as three distinct quantities, and to have no "relativistic mass"

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Even though I agree with Jerry Schirmer, I don't think he answered your question. I think you're asking how does GR simplify to SR (and more specifically how does mass in GR give the "relativistic mass" in SR)? Well, in GR we view spacetime as some curved manifold (think of it basically like some 4D shape). Importantly though, one can show that if you zoom in close enough (i.e look at a small enough patch of the manifold) spacetime looks flat. This is true everywhere, even in a black hole (you'd just have to zoom in a lot more). This is basically the mathematical way to formulate Einstein's equivalence principle. You can think of it like this: the earth is a sphere, but if you zoom in down to human size, it looks flat.

Once we've show this, then we can use all the technology from SR in this frame we just defined, including what you refer to as relativistic mass (but what Jerry rightly said is an outdated term). This is also what ensures that particle physics can be so accurate while ignoring gravity

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In general relativity we express the laws of physics using 4-vectors. 4-vectors are defined to have the same magnitude in all coordinate systems. E.g. the energy momentum 4-vector is $pc = (E, \mathbf p c )$. The magnitude of this vector is mass $mc^2$, where $$ m^2 c^4 = E^2 - \mathbf p^2c^2.$$ Mass is then an invariant quantity which does not increase with velocity. This is just as true in special relativity as in general relativity.

Unfortunately there is still confusion dating from the early days of special relativity, when momentum was still defined as a 3-vector $$\mathbf p = m\mathbf v. $$ using the velocity 3-vector. Writing relativistic formulae using 3-vectors caused people to define relativistic mass, which increases with velocity. We now recognise that relativistic mass is simply energy, and that it is much better to always use 4-vectors, and to drop the confusing idea that mass increases. There is no confusion about saying that energy increases with velocity.

Specifically we have the velocity 4-vector $$v= (\gamma , \gamma \mathbf v)$$ where $\gamma$ is the Lorentz factor, and we define the energy momentum 4-vector $$(E,\mathbf p) = p = mv= (\gamma m, \gamma m\mathbf v)$$ so, taking the 3-vector part, we have $$\mathbf p = \gamma m\mathbf v$$ which shows that it is really a mistake to define $\gamma m$ as relativistic mass, because the Lorentz factor $\gamma$ is associated with the kinematic factor $\mathbf v$, not with mass.

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