Hodge star operator on Minkowski spacetime
We spell out component expressions for the Hodge star operator on -dimensional Minkowski spacetime.
Conventions
We use Einstein summation convention throughout. With this convention, a generic differential p-form reads
Here denotes the factorial of .
We take the Minkowski metric to be the diagonal matrix of the form
We normalize the Levi-Civita symbol as
(1)
which means that
(2)
We normalize the sign of the volume form as
We write
(4)
for the generalized Kronecker delta, whose value is the signature of the permutation that takes the upper indices to the lower indices, if any such exists, and zero otherwise.
This appears whenever the Levi-Civita symbol is contracted with itself:
(5)
Notice the minus sign in (5), which comes, via (2), from the Minkowski signature.
Definition
We write for the operator of contraction of differential forms with the vector field , hence the linear operator on differential forms with anticommutator
With the volume form as in (3) it follows that (notice the reversion of the index ordering in the contraction operators )
Definition
For a differential p-form
its Hodge dual is:
(7)
where in the second line we used (6).
Properties
Proposition
(Hodge pairing)
For a differential p-form on -dimensional Minkowski spacetime its wedge product with its Hodge dual (7) is
(8)
Proof
We compute as follows:
Here the sign in the last lines arises from the Minkowski signature via (5).
Proposition
(double Hodge dual)
For a differential p-form on -dimensional Minkowski spacetime, its double Hodge dual (7) is
(9)
Proof
We compute as follows:
Here the sign in the last lines arises from the Minkowski signature via (5).
Proof
We compute as follows:
Here the sign in the last lines arises from the Minkowski signature via (5).
The views of space and time that I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality.
(Address to the 80th Assembly of German Natural Scientists and Physicians, (Sep 21, 1908), see WikiQuote)