This is the space of all smooth functions on $ M $ that vanish at $ \infty $. Let me make this more precise.

As $ M $ is a space-time manifold, there is an open cover $ \mathcal{U} $ of $ M $ and a $ \mathcal{U} $-sequence of embeddings $ \left( \phi_{U}: U \to \mathbb{R}^{4} \right)_{U \in \mathcal{U}} $ such that for all $ U,V \in \mathcal{U} $,
$$
\phi_{U} \circ \phi_{V}^{-1}: \quad
\mathbb{R}^{4} \supseteq {\phi_{V}}[U \cap V] \to
{\phi_{U}}[U \cap V] \subseteq \mathbb{R}^{4}
$$
is a smooth function between open subsets of $ \mathbb{R}^{4} $.

Now, to say that $ f \in {C_{0}}(M) $ means that

- $ f: M \to \mathbb{R} $;
- $ f \circ \phi_{U}^{-1}: \mathbb{R}^{4} \supseteq {\phi_{U}}[U] \to \mathbb{R} $ is a smooth function for each $ U \in \mathcal{U} $; and
- for any $ \epsilon > 0 $, there exists a compact subset $ K $ of $ M $ such that $ |f(x)| < \epsilon $ for all $ x \in M \setminus K $.