Why does raising indices on ε₀₁₂₃ give ε⁰¹²³ = -ε₀₁₂₃?

[binbagsss]

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

The soluton is:

ε$^{0123}$=g$^{00}$g$^{11}$g$^{22}$g$^{33}$ε$_{0123}$=-ε$_{0123}$

where g$_{\alpha\beta}$ is the metric tensor.

I am struggling to understand the last equality.

Many thanks for any assistance.

 

[Chestermiller]

g⁰⁰=-1, correct? Really, η⁰⁰=-1.