   Explaining Gravitational Time Dilation Geometrically

As general relativity is, at its core, a geometrical theory its effects can be demonstrated geometrically. To do so involves using objects called light cones. A light cone is an a cone drawn in space-time which represents the path taken by a time of light in a certain time. As one of Einstein showed in special relativity that the speed of light is the ultimate speed limit and nothing can exceed it everything else must travel within the arms of the light cone. In the above diagram the two diagonal red arrows represent where the path taken by the light travelling at speed c such that the distance travelled in time t is the same as the speed of light times by the time taken. The purple”worldline” represents a path taken by anything with mass (as it can’t travel faster or up to the speed of light) and can be anywhere between the two arms of the light cone.

In special relativity all the light cones line up in the same way as triangles line up on a flat plane and Pythagoras’ theorem holds everywhere. If, however the plane which the triangles are on Pythagoras’ theorem only holds true for each triangle in the individual position but they no longer line up with one another. The same applies to the light cones when gravity is introduced.

The curvature of space causes the light cones to tip over as they approach an object and the gravitational field strength (and thus the curvature of space-time) increases. The light cones no longer line, and when they reach rs from the object, are rotated such that there is no escape for the material as the only paths it can follow are towards the object, as shown in the diagram below

Diagram of a Light cone on space-time axis

As can be seen in the above diagram the light cones tip over as they approach the centre of mass and as worldlines must stay within the arms of the light when the cone has rotated through 90 degrees there is no choice but to travel in towards the centre.

If a message is sent out from something/someone falling towards a massive object at regular time intervals (either deliberately broadcasting something or just light reflecting off them) a person a long way away (ideally at infinity) would observe an increase in the time between each message. As the message would travel at the speed of light it would travel along the lines of the light cone.

However as the cones tip over each message dos not travel at the same angle as the last (they aren’t parallel) and therefore the messages diverge and so a person at infinity waits longer between each message. Ultimately at the event horizon the path the message takes is straight up the diagram (where the y axis represents time) so the observer has to wait infinitely long before the message reaches them. This is shown in the diagram below Diagram of the tipping of light cones as they approach a massive object

Whilst most objects that we encounter both in real life and the universe aren’t black holes the same principle applies, both the equation and the geometric representation. As an event happens closer to the earth the light cone tip overs more and thus for a distant observer time run slower. The only difference between the Earth and other objects we encounter in real life is that the mass isn’t contained within it’s Schwarzschild radius and so we can’t reach a point where the light cone lies on its side where time appears to stop from the observer’s view point Diagram of the slowing of time as an objet falls into a black hole

1. Ideas from Presentation 16 from the course Introduction to Astrophysics by Dr David Seery

2. Images are from the same presentation and our courtesy of Dr David Seery

3. Textbook “University Physics” not used