I think the answer depends on the metric signature (right?), so for this topic, let's define the signature of the metric to be positive(adsbygoogle = window.adsbygoogle || []).push({});

I'm curious what properties a tensor T must have, for the following to be true:

[tex]T^{ab} T_{ab} \ge 0[/tex]

Note: I am not talking about the stress energy tensor. I'm not sure what a good symbol choice is for "arbitrary tensor".

Similarly, I'm curious for higher rank tensors, and similar relations, like:

[tex]T^{abc} T_{abc} \ge 0[/tex]

and

[tex]T^{abcd} T_{abcd} \ge 0[/tex]

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For the rank 2 tensor, I can at least start an approach for this, but can't make it rigorous.

Choose a local inertial coordinate system, so the metric is diagonal (-1,1,1,... ). the coordinate components of T^ab will differ from T_ab only by a change in sign of the ((a=0 or b=0) and (a!=b)) components. For dimension N there will be 2(N-1) of these. However we have N(N-1)/2 arbitrary spacetime rotations we can perform and still be in a local inertial frame. So as long as

[tex]2(N-1) \le N(N-1)/2[/tex]

we can naively expect to have enough "constraint" to require that those components which change sign are all zero. And therefore, for dimension N>=4, and metric signature > 0, it looks like we can always garauntee

[tex]T^{ab} T_{ab} \ge 0[/tex].

But I think there are probably cases where this fails (the naive expectation that one can constrain certain components to be zero). Consider for instance the electromagnetic tensor. If there is an electric field with no magnetic field in one coordinate system, one can't find an inertial coordinate system in which there is no electric field.

But for symmetric tensors (as discussed in another thread), it should be possible to make it diagonal in a local inertial frame. So maybe the requirement is that the tensor is symmetric? Or is that overly restrictive? (or maybe even in that case there are problems as well?)

Is there an easier way to approach this?

And what conditions are necessary for the higher tank tensor cases? (For instance, does the Riemann curvature tensor have enough symmetry that [itex]R_{abcd}R^{abcd} \ge 0[/tex]?)

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# Is the tensor product T^ab T_ab non-negative?

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