There are some points in this [wikipedia][1] 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 $\rho$


  [1]: https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor#Stress%E2%80%93energy_of_a_fluid_in_equilibrium