General Relativity tensor proof

[regretfuljones]

Prove that if tauⁱ_jkl is a tensor such that, in the {xⁱ}-system, tauⁱ_jkl=3tauⁱ_ljk , then tauⁱ_jkl=3 tauⁱ_ljkin all coordinate systems.

How would one go about proving this for all coordinate systems?

 

[BLaH!]

Remember the similarity transformation of tensors. If I have a tensor that is given in the $$x_i$$ basis I can find what that tensor looks like in any other basis (say the $$x_j$$ basis) by the following

$$T^{x_j} = U^\dagger T^{x_i} U$$ (1)

Here, $$U$$ is a unitary tensor that relates the components of a vector described in the two different bases. The key property of $$U$$ is

$$U^\dagger U = U U^\dagger = I$$ (2)

Here, $$I$$ is the identity tensor. Anyway, play around with that. You won't need to actually calculate what $$U$$ is. You just have to know (1) and (2).

 

[wais]

The proof of this statement involves understanding the properties of tensors in general relativity and how they transform between different coordinate systems. In general relativity, tensors are mathematical objects that describe the geometric properties of spacetime and they are defined as quantities that transform in a specific way under coordinate transformations.

To prove that tauijkl=3 tauiljkin all coordinate systems, we first need to understand how tensors transform under coordinate transformations. In general, tensors transform according to the following rule:

T^{\alpha\beta\gamma...}_{\mu\nu\rho...} = \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} \frac{\partial x^{\gamma}}{\partial x'^{\rho}}... T'^{\mu\nu\rho...}_{\alpha\beta\gamma...}

Where T and T' represent the tensor in the original and transformed coordinate systems, respectively, and x and x' represent the coordinates in the original and transformed systems, respectively.

Now, let's apply this transformation rule to the given tensor tauijkl, where we have tauijkl=3tauiljk in the {xi}-system. We can write this as:

tauijkl = 3 tauijkl

Using the transformation rule, we can express this in the new coordinate system as:

tauijkl = \frac{\partial x^i}{\partial x'^j} \frac{\partial x^j}{\partial x'^k} \frac{\partial x^k}{\partial x'^l} 3 tauijkl

Since the transformation rule applies to all coordinate systems, this means that the above equation holds true for any coordinate system. Therefore, we can conclude that tauijkl=3tauiljkin all coordinate systems, as desired.

In summary, the proof of this statement relies on understanding the transformation properties of tensors in general relativity and applying the transformation rule to the given tensor. By showing that the equation holds true for any coordinate system, we can conclude that tauijkl=3 tauiljkin all coordinate systems.