Projection of a tensor Consider the following tensor (abstract index notation, e.g. Wald's) $B_{ab}$ and timelike vector field $X^{a}$ such that $X^aX_a=-1$ and 
\begin{equation}
B_{ab}=\nabla_bX_a
\end{equation}
Then one claims that $B_{ab}$ is purely spatial, i.e. $B_{ab}X^a=B_{ab}X^b=0$. I do not quite understand how this is interpreted as "spatial", though I presume it borrows the intuition that such operation is like dot product or projection (e.g. a vector is purely spatial with respect to timelike vector if it is orthogonal to the said timelike vector). [This example is from Wald chapter 9 section 9.2]
I have seen similar reasoning used in general relativity, e.g. when computing "transverse-traceless" component of a tensor. I can see "transverse" notion for waves, but to say a general tensor (even second rank) is transverse is still a bit confusing. Could anyone explain the concept physically? I would expect that the reason the terminology was borrowed from simpler physics is because they indeed have analogous interpretation.
 A: Take the covariant derivative of the equation $X^aX_a=-1$. The RHS becomes zero so we have $$2X_a\nabla_b X^a=0\implies X^aB_{ab}=0.$$
The other equation, $X^bB_{ab}=0$, is the geodesic equation, so it doesn't hold for just any $X^a$.
Let's consider the situation at some point $p\in M$. Then $X^a$ is a prime candidate for the timelike basis vector of $T_pM$. Let $E_1^a,E_2^a,E_3^a$ be a triple of orthonormal spacelike vectors in the subspace orthogonal to $T_pM$. Then $X^a,E_1^a,E_2^a,E_3^a$ forms a basis of $T_pM$. The two conditions means that the component expansion of $B^{ab}$ can only contain factors of $E_i^aE_j^b$, i.e. something like $X^aE_1^b$ is forbidden. It is in this sense that it is "purely spatial".
A: To add to Ocelo7's answer, the transverse part is, as I have seen it used by e.g. Ellis, used to refer to components that are orthogonal to a future-pointing and geodesic null vector field spanning the past light cones of a particular world line -- the central observer (e.g. our world line). They are defined on parts of spacetime that is observeable by the central observer, and they are taken to span the orthogonal (relative to the null vector) part of this space. This makes them 2-dimensional, but not in general orthogonal to the timelike vector field representing the velocity of the matter content.
For example the transverse velocity of the matter content describes the motion of distant objects transverse to the null geodesics connecting them to the central observer, e.g. how we observe the part of their motion relative to us that is not toward us nor away from us.
