2

I am reading Gravitation and Cosmology by Steven Weinberg. On page 133, he says

$R^{\lambda}_{\phantom{x}\mu\nu\kappa}$ is the only tensor that can be constructed from the metric tensor and its first and second derivatives, and is linear in the second derivatives.

I have never heard this. I'd like to read more about this property.

Can someone either (1) explain why this seems like a natural consequence given what we know about the Riemann tensor or (2) direct me to a resource so I can read about it myself?

Stan Shunpike
  • 5,293
  • 4
  • 43
  • 81
  • 1
    http://math.stackexchange.com/questions/578372/tensors-constructed-out-of-metric-other-than-the-riemann-curvature-tensor?rq=1 – Blah Feb 19 '15 at 06:52
  • You are unsatisfied with the proof on page 134? – Ryan Unger Feb 20 '15 at 21:25
  • No, I'm not dissatisfied. It makes sense and I see his line of reasoning. I just wondered if this is the only way to do it. But his way is clever and is how I would approach it. – Stan Shunpike Feb 20 '15 at 21:39
  • I'm always confused how much credit to give Einstein for coming up with general relativity. Its obviously genius, anyone can see that. But the question is would someone else during the time period have come up with it if he hadn't? And the fact that only one tensor can satisfy those properties is a pretty big clue...so I never would have expected a single tensors could satisfy the conditions Weinberg laid out. I was expecting a number of tensors could have been used. I was very surprised to learn the opposite. But no his proof is obviously perfectly satisfactory. I just was taken aback. – Stan Shunpike Feb 20 '15 at 21:45
  • Credit might be the wrong word. Its more like I am confused about measuring his historical impact compared with his peers and colleagues. – Stan Shunpike Feb 20 '15 at 21:47
  • @StanShunpike Einstein is usually given the sole credit for GR. (Much unlike SR, which was the product of many minds.) Hilbert is given credit for writing $\mathcal{L}=R$ and doing the variational problem and Grossmann can be given credit for teaching Einstein diffgeo. You see, the Eq. Principle was merely seen as a mathematical curiosity by his peers (the canceling of the $m$s in $ma=-m\phi'$). It was Einstein who became obsessed with geometry and he saw the connection. – Ryan Unger Feb 21 '15 at 03:35
  • Perhaps I am being silly, but...if $R$ satisfies these things, shouldn't $2R$ also? Indeed, if we add up terms, it would still be linear, so every contraction would also satisfy this, right? – Robin Goodfellow Feb 22 '15 at 13:40
  • Yes, Weinberg mentions the contraction part as well. But I still don't understand it. Perhaps you can elaborate? – Stan Shunpike Feb 22 '15 at 16:03
  • We can only determine a unique tensor up to the overall sign and scale. – Ryan Unger Feb 24 '15 at 21:06

0 Answers0