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In the Stress-Energy tensor of General Relativity, the momentum density in one direction can flow into another, which is an object's shear stress. Still, how can the momentum travelling in one direction flow into another? Thanks!

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  • $\begingroup$ because space-time is curved... Imagine a scenario in fewer dimensions $\endgroup$ – JMLCarter Sep 1 '17 at 14:35
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    $\begingroup$ Just imagine I am running by you while you are standing still on a skateboard. While I am approaching you, I push you or pull you a little bit. What will happen to you? You will move toward me, right? You get some momentum. This amount of momentum is the flow of the momentum. This flow of momentum passes from me to you, in the direction orthogonal to my momentum. Although I am running ahead, the force exerted on you transfers the momentum, and thus the flow. $\endgroup$ – Drake Marquis Sep 3 '17 at 4:12
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This has really nothing to do with general relativity or anything like that. It becomes much simpler if you remember that "flow of momentum" is another word for "force".

So consider this: you have a table on wheels. The top of the table is made of ice (so frictionless), except for one section which is regular wood and thus has friction. Sitting on the table on the icy part is a box.

You push the table. Initially the box stays in place, sliding over the ice since there is no friction to push it along. But when the box meets the wood, suddenly it is dragged along with the table thanks to friction. In effect, what you have done is transfer some $x$-momentum (i.e., horizontal) in the $y$ direction (i.e., vertically).

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Although you have asked in the context of general relativity, this also occurs in continuum mechanics in which stress has the same interpretation, i.e. as momentum flux. Momentum density oriented in one direction flowing into another direction is a consequence of the way we set up the coordinate system. Suppose you orient your coordinate system such that its X-axis points along the direction of momentum density. Then momentum density will have only one non-zero component which is the X-component, and its flux will also be in the X-direction, with no flux in other directions.

But momentum density vector has different directions at different points. So usually we fix a coordinate system with given direction for its axes. In that case direction of momentum density can have arbitrary orientation and in general is not aligned with any of the coordinate axes. It is then that we have a component of momentum density in a particular direction flowing into other directions.

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