Quick question, index notation, alternating tensor.

[binbagsss]

Q) I am using index notation to show that ε[itex]^{0123}[/itex]=-1 given that ε[itex]_{0123}[/itex]=1.

The soluton is:

ε[itex]^{0123}[/itex]=g[itex]^{00}[/itex]g[itex]^{11}[/itex]g[itex]^{22}[/itex]g[itex]^{33}[/itex]ε[itex]_{0123}[/itex]=-ε[itex]_{0123}[/itex]

where g[itex]_{\alpha\beta}[/itex] is the metric tensor.

I am struggling to understand the last equality.

Many thanks for any assistance.

 

[Dick]

Q) I am using index notation to show that ε[itex]^{0123}[/itex]=-1 given that ε[itex]_{0123}[/itex]=1.

The soluton is:

ε[itex]^{0123}[/itex]=g[itex]^{00}[/itex]g[itex]^{11}[/itex]g[itex]^{22}[/itex]g[itex]^{33}[/itex]ε[itex]_{0123}[/itex]=-ε[itex]_{0123}[/itex]

where g[itex]_{\alpha\beta}[/itex] is the metric tensor.

I am struggling to understand the last equality.

Many thanks for any assistance.

Look up the definition of the metric tensor you are using and insert the values of the g components.

 

[HallsofIvy]

If you are using the standard metric tensor for relativity then [itex]g^{11}= g^{22}= g{33}= -1[/itex] while [itex]g^{00}= 1[/itex].