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Not according to Richard Feynman:Does this mean it is just better to "shut the heck up and calculate"? That is, to forget about trying to "intuit" these things and just learn how to apply them mechanically to solve problems and generate predictions for experiments?

On the other hand such an understanding is not easy to come by.... he also says:"A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist."

“Do not keep saying to yourself, if you can possible avoid it, "But how can it be like that?" because you will go 'down the drain', into a blind alley from which nobody has escaped. Nobody knows how it can be like that.

pervect:

Interesting!: Einstein was guided in part via Lorentz, Fitzgerald and the 'ether' controversy of the early 1920's. I spent some time, unsuccessfully, trying to find out how the development of tensors in fluid mechanics may have influenced Einstein, how he might have gained insights from that work. I know there is documentation of Einstein's work because [I think it may have been Brian Greene who commented on it.] Einstein's original notes and his errors,false starts,corrections and alternative formulations were noted.I can also add that tensors have a long history of use before GR, particularly in fluid mechanics. Of course, the classical tensors transform via the Gallilean transformation - the relativistic tensors transform via the Lorentz transform.

Quantum Immortal:

I wanted to emphasize the following because I often forget.Is it correct to say...... The tensor represents, 'invariant energy'? I really want an answer to this one.

pervect:

This is a good way to 'intuit' that different observers see componets of the SET differently.There are several ways of talking about the stress energy tensor. One way of looking at it is this - the stress energy tensor, when multiplied by the 4-velocity of an observer, gives the purely spatial energy momentum density (which is a 4-vector) measured by an observer moving at that four velocity. This is the approach used by MTW in "Gravitation", basically

The

**components**of any tensor field, like the SET, only have meaning relative to a choice of frame. But the overall SET is a covariant object regardless of the fact that relative to a given frame field its components are not. That's the whole point of using tensors. Their components relative to a frame field correspond to frame dependent [say, varying observer velocity] locally measurable quantities but the tensors themselves are frame independent, geometric objects.

A single fast electron has the same 'gravitational spacetime curvature' as a slow one. But it follows a different trajectory. A moving object does not curve space-time any differently from a stationary object. [We take the SET gravitational source term in the frame of the object, or system.] So a 'hot' plasma has more 'gravity' than a cooler one because the faster moving mor energetic ions exert more pressure in the frame of the system. The overall KE may remain constant but the pressure does not.

The gravitational spacetime curvature is an intrinsic geometric quantity, and invariant under coordinate transformations. Moving objects do generate different apparent forces on test particles through the source term (stress-energy tensor) in the Einstein equation...so different trajectories result.

As an example of some of this, the [idealized] symmetrical collapse of a star does not change the external gravitational spacetime curvature unless one assumes radiation carries away some energy. The components of the SET varying in such as way as to maintain over all gravitational uniformity.

Quantum:

There is a nice complementary forum discussion here with some good physical insights:

Energy gravitates?

https://www.physicsforums.com/showthread.php?t=698100