Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 4; Problems 4.3, 4.4.
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.