Thinking General Relativity, 19

Anirudh Singh

7 min readMay 16, 2024

Timelike, Spacelike, and Light-like Intervals

The concepts of timelike, spacelike, and light-like intervals are fundamental for understanding the geometry of spacetime and the causal relationships between events.
These intervals are essential for characterizing the separation between two events in four-dimensional spacetime.

Timelike Interval

A timelike interval describes the separation between two events in spacetime for which there exists a possible causal connection, meaning that one event could influence the other without violating the speed limit of light.
Mathematically, the square of the timelike interval between two events is negative.
It’s expressed as:

The spacetime interval between two events along a timelike path is always measured by an observer to be less than zero, meaning that the “distance” between the two events is a timeline.
This indicates that the events are causally connected and that one event can influence the other.

Spacelike Interval

A spacelike interval describes the separation between two events in spacetime for which there is no possible causal connection.
Mathematically, the square of the spacelike interval between two events is positive.
It’s expressed as:

The spacetime interval between two events along a spacelike path is always measured by an observer to be greater than zero.
This indicates that the “distance” between the two events is spacelike, implying that no signal or influence can propagate between them.

Light-like Interval

A light-like interval describes the separation between two events in spacetime that are connected by a path along which a light signal could travel.
Mathematically, the square of the light-like interval between two events is zero. It’s expressed as:

The spacetime interval between two events along a light-like path is always measured by an observer to be equal to zero.
This indicates that the “distance” between the two events is light-like, meaning that they are connected by a path along which a light signal could propagate. Light-like intervals trace the trajectories of photons or light rays in spacetime.

Summarize and Visualize

Relation Between Events [2]

Events A and B form a timelike pair (with event A arbitrarily chosen as a reference event), here recorded in the spacetime maps of three free-float frames, Point B lies on a hyperbola opening along the time axis in each frame.
The shortest time between events A and B is recorded in the laboratory frame, the frame in which the two events occur at the same place.

The spacelike pair of events A and D {with event A arbitrarily chosen as the reference event) is recorded in the spacetime maps of three free-float frames.
Point D lies on a hyperbola opening along the space axis in every rocket and laboratory frame.
The shortest distance between these events is recorded in the laboratory frame, the frame in which the two events occur at the same time.
A heavy line represents the spacetime separation AD.
No particle can travel along this line; the speed would be greater than light speed — and would be infinitely great as measured in the laboratory frame since the particle would have to cover the distance from A to D in zero time!

Two lightlike pairs of events 𝐴𝐸 and 𝐴𝐺 (with event A arbitrarily chosen as reference event) as recorded in spacetime maps of three free-float frames.
A flash originates at A and spreads outward from the center of a rod at rest in the laboratory frame.
Events 𝐸 and 𝐺 are receptions of this flash at the two ends of the rod as recorded by different observers. In the laboratory frame, reception events 𝐸 and 𝐺 occur at the same time.
In the right-moving rocket frame, the rod moves to the left, so event 𝐺 occurs sooner than event 𝐸. In the left-moving rocket frame, the rod moves to the right, so event 𝐸 occurs sooner than event 𝐺.

The Lightcone Returns [4]

In attempting to diagram relativistic spacetimes, one of the most important features to capture is the causal structure of the spacetime.
This structure specifies which events (that is, which points of space and time) can be connected by trajectories that are slower than light, which events can be connected by trajectories traveling at the speed of light, and which events cannot be connected by anything traveling at or below light speed.
Events in the first group are said to be “timelike related” because a physical clock could travel from one event to the other. Events in the second group are “lightlike related” because a light ray can travel from one to the other.
Events in the third group are “spacelike related”. Given that it is physically impossible (on the standard interpretation of relativity theory) for any causal process to exceed the speed of light, these three possible ways of being connected tell us whether one event is able to influence another.
We can depict these three spatiotemporal relationships by drawing in the “light cone” of an event.
Given an event p, the light cone of p consists of all the points that can be connected to p by a straight ray of light.
Imagine that event p is someone flashing a bright light from a particular location. Then one second later, there will be a sphere of points that is currently occupied by the outgoing light pulse.
Two seconds later, a larger sphere, farther out, will be illuminated, and so on. We can depict these spheres at progressive times in a spacetime if we ignore one dimension of space so that we can draw in the points occupied by the light pulse as a circle, as in the following figure:

We can likewise depict the points in the past that are connected to point p by light rays.
For a given time, these points will again form a sphere, which will grow larger the farther into the past we look. Thus the full light cone looks like this:

We can now use these light cones to depict the causal structure of spacetime. Anything outside of the light cone of p cannot causally interact with p.
The “causal future” of p consists of the points on and inside the future half of the light cone.
Likewise, the causal past is picked out by the bottom half of the light cone.
Note that because nothing can go faster than light, the trajectory of any object will always remain within the light cone of each event along that trajectory — the path will “thread” the cones:

We depict flat spacetime by keeping all the light cones oriented in the same direction, as in the following figure.

The curvature of spacetime, then, will be depicted by the tilting of these light cones. This reflects the fact that the causal structure of such spacetimes is different from that of flat spacetimes.
So, for example, the spacetime around a massive body like a star will be depicted as follows.

We have a black hole when the curvature of spacetime becomes so severe that, for some region, there is no path out of that region that remains inside its own light cones.
That is, the causal structure of spacetime is such that one cannot escape from that region without traveling faster than light. Such a region is by definition a black hole; the border of that region is the event horizon.