**Required math: algebra**

**Required physics: special relativity**

Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 4; 3,4.

The invariant interval in special relativity can be written as

where is the metric tensor in flat space, with components , for and zero otherwise. Thus this relation is the same as

Under a Lorentz transformation, we get

Since the interval is invariant, we get

Since the last equation must be true for an infintesimal interval, the quantity in parentheses must be zero, so

That is, if we apply a Lorentz transformation (the *same* transformation!) to each index in the metric tensor, we get the same tensor back again.

We can multiply this equation by an inverse transformation to get

Multiplying a transformation by its inverse gives the identity matrix:

So we get

Repeating the process, we get

Thus, not surprisingly, if we multiply the metric tensor by two inverse Lorentz transformations, we get the same tensor back.

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