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I'm dealing with the following:

$$\eta^{\alpha \mu} \eta_{\alpha \nu} \phi,_{\beta \mu}$$

$$\eta^{\alpha \beta} \phi,_{\alpha \beta}$$

where $\eta$ is the Minkowski metric and $\phi$ is a function of the spacetime coordinates.

I know the first expression above turns simply into $\phi, _{\beta \nu}$ and the second to $\phi,_{\alpha}^{,\alpha}$but I am unsure how. I generally know how Einstein summation convention works, but an explicit illustration with this example would be helpful.

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  • $\begingroup$ This is special relativity, not general relativity. $\endgroup$ – Ben Crowell Nov 9 at 22:47
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    $\begingroup$ Contracting any index of any tensor with one of the two indices of the metric tensor raises or lowers the original index and changes it to the other index in the metric tensor. $\endgroup$ – G. Smith Nov 9 at 22:48
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    $\begingroup$ Let $T_{\mu \nu}$ be a second order tensor. Suppose I want to raise both indices on $T_{\mu \nu}$. Then $\eta^{\alpha \mu} T_{\mu \nu}\eta^{\nu \beta}=T^{\alpha \beta}$. Suppose I needed to convert $T_{\mu \nu}$ to $T_{\mu}^{\; \nu}$. Then $T_{\mu \nu}\eta^{\nu \beta}=T_{\mu}^{\; \beta}$. $\endgroup$ – Cinaed Simson Nov 10 at 0:06
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Let's say that you have a tensor $T{^m}_n $ and you want to contract the lower index. You can do this simply by multiplying with another tenso $L^{nk}$ and you will have ${T{^m}_n} L^{nk}=J^{mk} $. Also, in special relativity the Lorentz transformation can be written as $$x^{'\mu} ={\Lambda}{^\mu} _{\nu}x^{\nu} \ \ \ \ \ \ {\Lambda } {^\mu}_{\rho} {\eta}_{{\mu}{\nu} } {\Lambda }{^\nu}_{\lambda} ={\eta} _{{\rho}{\lambda} } $$When you have a repeating index, this means that you sum over this index. This is the Einstein summation convention.

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