Hi,
I really don't understand Einstein summation convention. I'm working with Hans Ohanian, Gravitation and Spacetime. I'm told that "A tensor of rank r is an object \(\displaystyle $A^{\mu\nu...\kappa}$\) with \(\displaystyle $4^r$\) components that under a Lorentz transformation transforms according to \(\displaystyle $A'^{\alpha\beta...\gamma}=a^\alpha_\mu a^\beta_\nu...a^\gamma_\kappa A^{\mu\nu...\kappa}$\)" where \(\displaystyle ${a^\mu_\nu}$\) is the Lorentz transformation matrix.
I'm then told to write "the 16-component object \(\displaystyle $x^\mu x^\nu$\)" as a matrix and prove that it transforms as a second rank tensor.
I have no idea how to do this. The upper index means they're column vectors, right? so how does it make sense to combine them? I tried writing it out as a four-by-four matrix with each of the terms multiplied, which kind of makes sense to me, except that when I multiply that by two Lorentz transformation matrices I do NOT get the basis-changed matrix I'm looking for.
I've been stuck on this for ages, and I would really appreciate some help. Thanks.
I really don't understand Einstein summation convention. I'm working with Hans Ohanian, Gravitation and Spacetime. I'm told that "A tensor of rank r is an object \(\displaystyle $A^{\mu\nu...\kappa}$\) with \(\displaystyle $4^r$\) components that under a Lorentz transformation transforms according to \(\displaystyle $A'^{\alpha\beta...\gamma}=a^\alpha_\mu a^\beta_\nu...a^\gamma_\kappa A^{\mu\nu...\kappa}$\)" where \(\displaystyle ${a^\mu_\nu}$\) is the Lorentz transformation matrix.
I'm then told to write "the 16-component object \(\displaystyle $x^\mu x^\nu$\)" as a matrix and prove that it transforms as a second rank tensor.
I have no idea how to do this. The upper index means they're column vectors, right? so how does it make sense to combine them? I tried writing it out as a four-by-four matrix with each of the terms multiplied, which kind of makes sense to me, except that when I multiply that by two Lorentz transformation matrices I do NOT get the basis-changed matrix I'm looking for.
I've been stuck on this for ages, and I would really appreciate some help. Thanks.