- #1
illuminates
- 26
- 0
I set eyes on the next formulas:
\begin{align}
E_{\alpha \beta \gamma \delta} E_{\rho \sigma \mu \nu} &\equiv g_{\alpha \zeta} g_{\beta \eta} g_{\gamma \theta} g_{\delta \iota} \delta^{\zeta \eta \theta \iota}_{\rho \sigma \mu \nu} \\
E^{\alpha \beta \gamma \delta} E^{\rho \sigma \mu \nu} &\equiv g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} \delta^{\rho \sigma \mu \nu}_{\zeta \eta \theta \iota} \\
E^{\alpha \beta \gamma \delta} E_{\rho \beta \gamma \delta} &\equiv -6 \delta^{\alpha}_{\rho} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \gamma \delta} &\equiv -2 \delta^{\alpha \beta}_{\rho \sigma} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \theta \delta} &\equiv -\delta^{\alpha \beta \gamma}_{\rho \sigma \theta} \,.
\end{align}
in https://en.wikipedia.org/wiki/Levi-Civita_symbol#Levi-Civita_tensors I want to know for what signature (+---) or (-+++) it is given. Is there simple way to check signature?
In my opinion there are need plus rather than minus at least in last three formulas:
\begin{align}
E^{\alpha \beta \gamma \delta} E_{\rho \beta \gamma \delta} &\equiv 6 \delta^{\alpha}_{\rho} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \gamma \delta} &\equiv 2 \delta^{\alpha \beta}_{\rho \sigma} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \theta \delta} &\equiv \delta^{\alpha \beta \gamma}_{\rho \sigma \theta} \,.
\end{align}
\begin{align}
E_{\alpha \beta \gamma \delta} E_{\rho \sigma \mu \nu} &\equiv g_{\alpha \zeta} g_{\beta \eta} g_{\gamma \theta} g_{\delta \iota} \delta^{\zeta \eta \theta \iota}_{\rho \sigma \mu \nu} \\
E^{\alpha \beta \gamma \delta} E^{\rho \sigma \mu \nu} &\equiv g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} \delta^{\rho \sigma \mu \nu}_{\zeta \eta \theta \iota} \\
E^{\alpha \beta \gamma \delta} E_{\rho \beta \gamma \delta} &\equiv -6 \delta^{\alpha}_{\rho} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \gamma \delta} &\equiv -2 \delta^{\alpha \beta}_{\rho \sigma} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \theta \delta} &\equiv -\delta^{\alpha \beta \gamma}_{\rho \sigma \theta} \,.
\end{align}
in https://en.wikipedia.org/wiki/Levi-Civita_symbol#Levi-Civita_tensors I want to know for what signature (+---) or (-+++) it is given. Is there simple way to check signature?
In my opinion there are need plus rather than minus at least in last three formulas:
\begin{align}
E^{\alpha \beta \gamma \delta} E_{\rho \beta \gamma \delta} &\equiv 6 \delta^{\alpha}_{\rho} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \gamma \delta} &\equiv 2 \delta^{\alpha \beta}_{\rho \sigma} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \theta \delta} &\equiv \delta^{\alpha \beta \gamma}_{\rho \sigma \theta} \,.
\end{align}