# Why are shear-stress and momentum-flux the same in the GR?

I am investigeting the meaning of the components of the Stress-Energy tensor:

My source also states, that this matrix is always symmetric in the General Relativity. That looks obvious on the image - except the pair of the momentum flux and shear stress. On this picture, these quantities are the same.

Why would be momentum flux the same as the shear stress? These look completely unrelated for me.

• The units of shear stress (Force/Area) are the same as the units of momentum flux. The wikipedia page for momentum diffusion gives a nice explanation of this. Nov 13, 2020 at 13:52
• @Framazu THat is okay but, for example, also the units of the pressure and the energy density are the same while they are very different things. Nov 13, 2020 at 14:38

To keep things simple, assume $$x^i$$ are Cartesian coordinates in flat spacetime. Let's say we have a particle of mass $$m$$ with velocity $$(v^1,v^2,v^3)$$ and momentum $$\gamma m (v^1,v^2,v^3)$$. Now, let's focus on the component $$T^{ij}$$. It represents the flux of the $$i$$-th component of momentum through a surface of constant $$x^j$$. Mathematically it would be $$\gamma mv^i (\text{d}x^j/\text{d}t) = \gamma mv^iv^j$$. Notice that this is symmetric in $$v^i$$ and $$v^j$$ so it also represents $$T^{ji}$$. So we must have $$T^{ij}=T^{ji}$$.

Here's an explanation without math. Again, let's imagine a Newtonian particle with some momentum each in the $$x$$ and $$y$$ directions. If we double the $$x$$ component of momentum, the particle is now carrying $$y$$-momentum twice as fast in the $$x$$ direction. So $$T^{yx}$$ doubles. However at the same time it is carrying twice as much $$x$$-momentum in the $$y$$ direction. So $$T^{xy}$$ also doubles.

I hope this is enough to convince you that the 2 quantities are indeed equal. Including relativistic corrections into the second paragraph would just make the $$x$$-velocity increase by less than a factor of $$2$$, so the idea remains unchanged.