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There are some points in this wikipedia chapter. Main equation is:

$$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$

where $c$ is explicit.

The one for the trace is:

$$T = 3p - \rho c^2$$

that seems contradictory with:

$$T^{\alpha\beta} = \left( \begin{matrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$$

with trace $3p+\rho$ (difference in sign and value of last term).

The expression for the four-velocity:

$$u^{\alpha} = (1, 0, 0, 0)$$

is not the usual one $(c, 0, 0, 0)$.

Finally, the metric:

$$g^{\alpha\beta} \, = \left( \begin{matrix} - c^{-2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

also with explicit $c$, it is also not the usual:

$$\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

Are the wikipedia equations in this chapter using a coherent notation ? If yes, how to explain the previous points ?

Addendum:

After the good answer from @MikeStone about the trace, one point seems still open. Starting from the first expression of stress tensor and replacing the values given in wiki for the four-velocity and metric, we obtain for the first row and column component:

$$ T^{0 0} \, = \left(\rho + {p \over c^2}\right)u^0u^0 + p g^{0 0} = \left(\rho + {p \over c^2}\right) \cdot 1 \cdot 1 + p \left( -c^2 \right) = \rho + {p \over c^2} -c^2p $$

that differs from the expected $\rho$

There are some points in this wikipedia chapter. Main equation is:

$$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$

where $c$ is explicit.

The one for the trace is:

$$T = 3p - \rho c^2$$

that seems contradictory with:

$$T^{\alpha\beta} = \left( \begin{matrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$$

with trace $3p+\rho$ (difference in sign and value of last term).

The expression for the four-velocity:

$$u^{\alpha} = (1, 0, 0, 0)$$

is not the usual one $(c, 0, 0, 0)$.

Finally, the metric:

$$g^{\alpha\beta} \, = \left( \begin{matrix} - c^{-2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

also with explicit $c$, it is also not the usual:

$$\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

Are the wikipedia equations in this chapter using a coherent notation ? If yes, how to explain the previous points ?

There are some points in this wikipedia chapter. Main equation is:

$$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$

where $c$ is explicit.

The one for the trace is:

$$T = 3p - \rho c^2$$

that seems contradictory with:

$$T^{\alpha\beta} = \left( \begin{matrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$$

with trace $3p+\rho$ (difference in sign and value of last term).

The expression for the four-velocity:

$$u^{\alpha} = (1, 0, 0, 0)$$

is not the usual one $(c, 0, 0, 0)$.

Finally, the metric:

$$g^{\alpha\beta} \, = \left( \begin{matrix} - c^{-2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

also with explicit $c$, it is also not the usual:

$$\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

Are the wikipedia equations in this chapter using a coherent notation ? If yes, how to explain the previous points ?

Addendum:

After the good answer from @MikeStone about the trace, one point seems still open. Using first equation of stress tensor and replacing the values given in wiki for the four-velocity and metric, we obtain for the first row and column:

$$ T^{0 0} \, = \left(\rho + {p \over c^2}\right)u^0u^0 + p g^{0 0} = \left(\rho + {p \over c^2}\right) \cdot 1 \cdot 1 + p \left( -c^2 \right) = \rho + {p \over c^2} -c^2p $$

that differs from the expected $\rho$

There are some points in this wikipedia chapter. Main equation is:

$$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$

where $c$ is explicit.

The one for the trace is:

$$T = 3p - \rho c^2$$

that seems contradictory with:

$$T^{\alpha\beta} = \left( \begin{matrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$$

with trace $3p+\rho$ (difference in sign and value of last term).

The expression for the four-velocity:

$$u^{\alpha} = (1, 0, 0, 0)$$

is not the usual one $(c, 0, 0, 0)$.

Finally, the metric:

$$g^{\alpha\beta} \, = \left( \begin{matrix} - c^{-2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

also with explicit $c$, it is also not the usual:

$$\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

Are the wikipedia equations in this chapter using a coherent notation ? If yes, how to explain the previous points ?

Addendum:

After the good answer from @MikeStone about the trace, one point seems still open. Starting from the first expression of stress tensor and replacing the values given in wiki for the four-velocity and metric, we obtain for the first row and column component:

$$ T^{0 0} \, = \left(\rho + {p \over c^2}\right)u^0u^0 + p g^{0 0} = \left(\rho + {p \over c^2}\right) \cdot 1 \cdot 1 + p \left( -c^2 \right) = \rho + {p \over c^2} -c^2p $$

that differs from the expected $\rho$

There are some points in this wikipedia chapter. Main equation is:

$$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$

where $c$ is explicit.

The one for the trace is:

$$T = 3p - \rho c^2$$

that seems contradictory with:

$$T^{\alpha\beta} = \left( \begin{matrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$$

with trace $3p+\rho$ (difference in sign and value of last term).

The expression for the four-velocity:

$$u^{\alpha} = (1, 0, 0, 0)$$

is not the usual one $(c, 0, 0, 0)$.

Finally, the metric:

$$g^{\alpha\beta} \, = \left( \begin{matrix} - c^{-2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

also with explicit $c$, it is also not the usual:

$$\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

Are the wikipedia equations in this chapter using a coherent notation ? If yes, how to explain the previous points ?

Addendum:

After the good answer of @MikeStone, only one point seems open. Using first equation of stress tensor and replacing the values given in wiki for the four-velocity and metric, we obtain for the first row and column:

$$ T^{0 0} \, = \left(\rho + {p \over c^2}\right)u^0u^0 + p g^{0 0} = \left(\rho + {p \over c^2}\right) \cdot 1 \cdot 1 + p \left( -c^2 \right) = \rho + {p \over c^2} -c^2p $$

that differs from the expected $\rho$

There are some points in this wikipedia chapter. Main equation is:

$$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$

where $c$ is explicit.

The one for the trace is:

$$T = 3p - \rho c^2$$

that seems contradictory with:

$$T^{\alpha\beta} = \left( \begin{matrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$$

with trace $3p+\rho$ (difference in sign and value of last term).

The expression for the four-velocity:

$$u^{\alpha} = (1, 0, 0, 0)$$

is not the usual one $(c, 0, 0, 0)$.

Finally, the metric:

$$g^{\alpha\beta} \, = \left( \begin{matrix} - c^{-2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

also with explicit $c$, it is also not the usual:

$$\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)$$

Are the wikipedia equations in this chapter using a coherent notation ? If yes, how to explain the previous points ?

Addendum:

After the good answer from @MikeStone about the trace, one point seems still open. Using first equation of stress tensor and replacing the values given in wiki for the four-velocity and metric, we obtain for the first row and column:

$$ T^{0 0} \, = \left(\rho + {p \over c^2}\right)u^0u^0 + p g^{0 0} = \left(\rho + {p \over c^2}\right) \cdot 1 \cdot 1 + p \left( -c^2 \right) = \rho + {p \over c^2} -c^2p $$

that differs from the expected $\rho$