Relativistic spacetime

A relativistic spacetime is a tuple $(M, O, A, \nabla, g, T)$ .

It consists of a 4-dimensional topological manifold $(M, O, A)$ carrying a torsion-free linear connection $\nabla$ , which is compatible with a Lorentzian metric $g$ (making it a Lorentzian manifold). Crucially, it must also be equipped with a time orientation $T$ .
Note: signature $(+, -, -, -)$ .

Motivation: mathematical geodesics vs. physical worldlines

Why do we need this extra "Time Orientation" structure?
Consider the analogy with curved surfaces (Riemannian manifolds). On a surface, any tangent vector at any point generates a valid geodesic. Even the zero vector generates a valid "constant curve" geodesic ( $γ (t) = p$ ).
In spacetime, the mathematical definition of a geodesic ( $\nabla_{\dot{γ}} \dot{γ} = 0$ ) remains the same, but the physical interpretation changes.

The "Rest" Problem: A constant curve $γ (t) = p$ is a mathematical geodesic in spacetime, but it represents a particle existing for zero duration. A physical particle "at rest" must actually move through time.
Causality: The metric allows for "spacelike" curves (moving faster than light), which satisfy the geodesic equation but are physically forbidden for massive bodies.

To extract physical worldlines from the set of all mathematical geodesics, we must impose restrictions: the tangent vector must be non-zero, causal (timelike or null), and future-directed. The metric $g$ alone defines causality, but it cannot define "future-directed." For that, we need a Time Orientation.

Time Orientation

The absolute time function in Newtonian spacetime associates to each $p \in M$ a time. That is, given any point you can just quote the time of that point unarguably. We used the absolute time function to define a future-directed vector field $X$ as $i_{X} d t > 0$ . Pictorially, this is given by an arrow pointing to the "upper side" of a tangent plane to a constant $t$ surface.

\begin{document}
    \begin{tikzpicture}
        \draw[ultra thick] (0,0) .. controls (3,1) and (5,-1) .. (7,0.8);
        \draw[ultra thick] (0,1.5) .. controls (3,2.5) and (5,0.5) .. (7,2.3);
        \node at (7.5,0.8) {\Large{$t_1$}};
        \node at (7.5,2.3) {\Large{$t_2$}};
        \draw[ultra thick, blue, rotate around={10:(1,0.23)}] (-0.5,0.23) -- (2.5,0.23);
        \draw[->, ultra thick, red] (1,0.23) -- (1.5, 1.5);
        \node[circle, fill, inner sep=1.5pt, label={below:\Large{$p$}}] at (1,0.23) {};
        \node at (0.8,1) {\color{red}\Large{$X$}};
        \node at (-0.8, 0.2) {\color{blue}\Large{$dt$}};
    \end{tikzpicture}
\end{document}

In Relativistic Spacetime, there is no absolute time function. The metric $g$ partitions vectors into three classes, but it is symmetric ( $v$ and $- v$ have the same norm):

Timelike: $g (v, v) > 0$ (inside the cones)
Lightlike (Null): $g (v, v) = 0$ (on the cones)
Spacelike: $g (v, v) < 0$ (outside the cones)

\begin{document}
    \begin{tikzpicture}
  % Draw light cone
\draw[thick, dashed] (0,0) -- (3,3);  % Future light cone
\draw[thick, dashed] (0,0) -- (-3,3);
\draw[thick, dashed] (0,0) -- (3,-3); % Past light cone
\draw[thick, dashed] (0,0) -- (-3,-3);

% Draw the time axis
\draw[->] (0,-3.5) -- (0,3.5) node[anchor=south] {$t$};

% Draw the space axis
\draw[->] (-3.5,0) -- (3.5,0) node[anchor=west] {$x$};

% Timelike vector (within the light cone)
\draw[->, thick, blue] (0,0) -- (1,2) node[midway, above, sloped] {Timelike vector};

% Spacelike vector (outside the light cone)
\draw[->, thick, red] (0,0) -- (2,1) node[midway, above, sloped] {Spacelike vector};

% Lightlike vector (on the light cone)
\draw[->, thick, green] (0,0) -- (1.5,1.5) node[midway, above, sloped] {Lightlike vector};

% Labels for regions
\node at (1.5,2.5) {Future Light Cone};
\node at (1.5,-2.5) {Past Light Cone};
\node at (2.5,0.5) {Spacelike Region};  
    \end{tikzpicture}
\end{document}

So we need some other way to define what a future-directed vector field is. First, we need to restrict to Lorentzian manifolds which are oriented, $(M, O, A^{↑}, g)$ , and define:
Definition (Time Orientation).
Let $(M, O, A^{↑}, g)$ be an oriented Lorentzian manifold. Then a time orientation is given by a smooth vector field $T$ that

does not vanish anywhere, and
$g (T, T) > 0$ .
$◼$

This field $T$ selects one lobe of the light cone at every point $p$ as the "future cone." Because $T$ is smooth, the choice of future varies continuously across the manifold—the future direction doesn't suddenly "flip" as you move from $p$ to $q$ .
With this, we can finally define a worldline strictly: It is a geodesic $γ$ such that $\dot{γ}$ lies in the same cone as $T$ (future-directed) and is timelike (for massive particles) or null (for light).

\begin{document}
    \begin{tikzpicture}[scale=1.3]
        \draw[draw=red, opacity=0.2, fill=red, fill opacity=0.2] (0,2) -- (-0.2,3.4) .. controls (2,4) and (3,1) .. (5,1.5) -- (4,0) .. controls (3,-0.5) and (2,3) .. (0,2);
        \node at (-0.5,3.5) {\color{red}\Large{$T_p$}};
        \node at (5.2,1.6) {\color{red}\Large{$T_q$}};
        \node at (2.2,2.1) {\color{red}\Large{$T$}};
        \draw [thick](-1,3) arc (180:0:1cm and 0.15cm);
        \draw[->,ultra thick,red] (0,2) -- (-0.2,3.4);
        \draw [thick](-1,3) arc (180:360:1cm and 0.15cm) -- (0,2) -- cycle;
        \draw [dashed,thick](-1,1) arc (180:360:1cm and 0.15cm) -- (0,2) -- cycle;
        \draw [dashed,thick](-1,1) arc (180:0:1cm and 0.15cm);
        \draw [thick,rotate around={-30:(4,0)}](3.5,1.5) arc (180:0:0.5cm and 0.1cm);
        \draw[->,ultra thick,red] (4,0) -- (5,1.5);
        \draw [thick,rotate around={-30:(4,0)}](3.5,1.5) arc (180:360:0.5cm and 0.1cm) -- (4,0) -- cycle;
        \draw [dashed,thick,rotate around={-30:(4,0)}](3.5,-1.5) arc (180:360:0.5cm and 0.1cm) -- (4,0) -- cycle;
        \draw [dashed,thick,rotate around={-30:(4,0)}](3.5,-1.5) arc (180:0:0.5cm and 0.1cm);
        \node[circle, fill, inner sep=1.5pt, label={left:\Large{$p \,\,$}}] at (0,2) {};
        \node[circle, fill, inner sep=1.5pt, label={right:\Large{$q$}}] at (4,0) {};
    \end{tikzpicture}
\end{document}

Note: We shall now simply refer to relativistic spacetime as just spacetime.

Stationary spacetime

Definition (Stationary Spacetime). A spacetime $(M, O, A, g, T)$ is called stationary if it admits a Killing vector field $K$ such that $g (K, K) > 0$ .

In a general spacetime, the time orientation $T$ is an arbitrary choice we impose to separate future from past. However, in a stationary spacetime, the geometry itself possesses a "time symmetry." The timelike Killing field $K$ provides a canonical time orientation. Since $K$ is timelike and smooth, it naturally singles out a future direction that respects the symmetries of the metric.
Claim A stationary spacetime is one where we can find a chart such that the components of the metric do not depend on time.
Proof. Recall a vector field is Killing if $L_{K} g = 0$ . The exercise at the end of lecture 11 (Schuller GR lectures) shows that in a chart this condition reads

T^{c} g_{a b, c} + g_{c b} T^{c}_{, a} + g_{c a} T^{c}_{, b} = 0.

Now imagine we pick a chart such that $T = δ_{0}^{a} \partial_{a} = \partial_{0}$ , then the second two terms vanish and we are simply left with

g_{a b, 0} = 0,

which is the statement that the metric components are time-independent in this chart. □

Worked example

high school gravity in GR