Computing the non-zero, independent components of the Riemann Tensor
The Riemann curvature tensor is a rank four tensor and it is used to describe space-time. has some peculiar properties i.e
i.e it is anti-symmetric in the pairs and but it is symmetric upon the interchange of the pairs and .
which is referred to as the first Bianchi Identity.
Using the above constraint equation, assuming that can take values between and , one can compute that the total number of non-zero and independent components of the Riemann tensor are . In (3+1) space-time, there are twenty, in (2+1) space-time dimensions, there are 6 and in (1+1) dimensions, there is only one.
Now that I have introduced what the Riemann tensor and it’s properties are, my question is how does one go about computing these independent, non-zero components i.e what are the different combinations of a,b,c,d!
Naively thinking, one can start with
for a = 0,3 for b = 0,3 for c = 0,3 for d = 0,3 if a != b if c != d print a,b,c,d
to get the combinations of a,b,c,d for which we get a non-zero Riemann tensor. But the problem now is that we have a lot of redundancy i.e
I am not able to figure out how to weed out the dependent components of the Riemann tensor and get only the independent ones. If you have a solution, hit me because I haven’t been able to find one online. So far…
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