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Momentum density in the $x$, $y$ and $z$ directions are components of the Stress-Energy Tensor in Einstein's Field Equations. Yet, how did Einstein come up with that conclusion? Is it possible to demonstrate why momentum should curve spacetime without as much mathematics? Also, why do the $T_{01}$, $T_{02}$, $T_{03}$, $T_{10}$, $T_{20}$ and $T_{30}$ components of the Stress-Energy Tensor refer to momentum density? I'd love to read an intuitive explanation to why the flow of energy from the time dimension to the $x$ direction, for example, represents momentum. Thanks!

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Momentum occurs in the first row as part of the way that the stress-energy tensor expresses $f=ma$. The space derivative of stress (the other three rows) gives $f$. The time derivative of momentum gives $ma$. We get $f=ma$ when the sum of the corresponding derivatives add to zero.

The goal of early GR was to find out how to use special relativity in the context of gravitational fields and accelerating systems. A breakthrough was the assumption that mechanics and electricity should follow the same laws in any falling reference frame (EEP). In particular, the sum of the appropriate derivatives of the stress-energy tensor should add to zero. This doesn't quite work out in falling reference frames because of tidal forces. This is remedied by finding a space-time surface where the sum of the covariant derivatives do add to zero. The Einstein Field Equations tell you how to do this, for an arrangement of matter-energy in motion, given by the stress-energy tensor. These equations must hold if we are to accept EEP.

We can hand-wave over the effect of momentum on gravity in extreme cases, by noting the relationships between momentum, energy, and mass.

The first column of the stress-energy tensor is used to expresses conservation of mass-energy. Think of momentum as matter flux. The space derivative of momentum gives the rate of change mass-energy. Stating that the sum of appropriate derivatives is zero, means mass-energy is conserved.

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