Could someone explain me what the parenthesis with tensors included mean? The symmetric part of a tensor is denoted using parentheses as:
$$T_{(ab)}=\dfrac{1}{2}(T_{ab}+T_{ba})$$  
And antisymmetric as:  
$$T_{[ab]}=\dfrac{1}{2}(T_{ab}-T_{ba})$$
With that in mind, here is a Tensor obtained from General relativity and Gravitation.
$$ \dfrac{1}{2} R_{abcd}u^d=\omega_{c[a;b]}+\sigma_{c[a;b]}+\dfrac{1}{3}h_{c[a}\Theta_{,b]} -\dot{u}_{c;[b}u_{a]}+....$$
Note that the last factor of the equation; $\dot{u}_{c;[b}u_{a]}$ can't be explained with those in mind. So there is something basic I'm missing?
 A: Define $T_{cba} = \dot{u}_{c;b}u_{a}$, then this is simply $T_{c[ba]}$, which is to say $(T_{cba}-T_{cab})/2$, which is $(\dot{u}_{c;b}u_{a} - \dot{u}_{c;a}u_{b})/2$
A: May I suggest the following (i) starting from the definition of the Riemann tensor contract it with the vector ${\bf u}$ whose components are $u^{d}$, which I assume is the tangent vector to a timelike geodesic in a timelike geodesic congruence (ii) use the definitions of the shear tensor $\sigma_{ca}$, the rotation 2-form $\omega_{ca}$, the expansion $\Theta$, the projection tensor $h_{ca}$, and $\dot{u}_c$, which I assume means 
$$
\dot{u}_c = u_{c;n}u^{n}\,,
$$
and then compare both sides. 
I would also point out that in Exercise 22.6 of ``Gravitation" by Misner, Thorne and Wheeler one is asked to show, for a field of fluid 4-velocities, that 
$$ u_{c;a} = \omega_{ca} + \sigma_{ca} + \frac{1}{3}\Theta h_{ca} - a_{c}u_{a}\,
$$
where $a_{c} = u_{c;n}u^{n}\,$ is the four-acceleration of the fluid, $\Theta$ is the ``expansion" of the fluid world lines $\Theta = u^{c}_{\,;c}$, $h_{ca}$ is the projection tensor given by
$$
h_{ca}\equiv g_{ca} + u_{c}u_{a}\,,
$$
$\sigma_{ca}$ is the shear tensor of the fluid given by
$$
\sigma_{ca} \equiv \frac{1}{2}(u_{c;n}h^{n}\,_{a} + u_{a;n}h^{n}\,_{c}) - \frac{1}{3}\Theta\,h_{ca}
$$
and $\omega_{ca}$ is the rotation 2-form of the fluid defined as
$$
\omega_{ca}\equiv\,\frac{1}{2}(u_{c;n}h^{n}\,_{a} - u_{a;n}h^{n}\,_{c})\,.
$$
Based on this, calculating the covariant derivative of $u_{c;a}$, i.e. $u_{c;a;b}$ and then forming $u_{c;b;a}$ and finding their difference would be a good place to start, since 
$$
u_{c;b;a} - u_{c;a;b} = R_{abcd}u^{d}\,.
$$ 
