Concept of time in General Relativity

I am starting to read General Relativity, and from what i have understood, in this theory, time is a derived notion. An event is not associated with a particular time.

But then what is the physical meaning of an event?

And if time is a derived notion, then the four dimensions in our manifold are all spatial?

• The Wikipedia article on coordinate time may be helpful. Dec 2 '18 at 1:02
• Time is derived in this sense means that you are free to do general coordinate transformations, which is the symmetry of GR; so there is no canonical meaning/definition of time. But time is still 'different' from space if the signature of the metric is $-1$ Dec 3 '18 at 14:52
• @PM2Ring Does time's negative sign, in GR, imply that a large spatial curvature at the point of a spacetime event tends to be accompanied by a small temporal curvature, and vice-versa? (Please excuse my ignorance, but I've never studied non-Riemannian geometry: Consequently, I'm using "large spatial curvature" to mean "curvature that, although perhaps varying, is on average continuous in some spatial direction", and "small temporal curvature" to mean "curvature that, although perhaps varying, is on average continuous in a temporal direction". I hope I'm not drawing too much from SR diagrams.) Jul 20 '19 at 18:36
• @Edouard You should ask about that in a separate question, not in the comments of someone else's question. Jul 20 '19 at 18:40
• You're right: I just hadn't realized that the ban on my turgid and convoluted questions had expired. However, in case anyone else has a similar question while these comments are still showing, John Rennie provided a very down-to-earth answer in his response to the question, "Is acceleration caused by curvature or space or time or both?" Jul 20 '19 at 19:47

You are correct in saying a particular event is not associated with a particular time.

This does not mean that time is derived, though. It's just different for observers travelling at different speeds.

An event is just a point in spacetime. Usually, we care about interesting events, like a lightening strike, or an explosion, or a train passing a flare or whatever. The catch is that observers travelling at different speeds will measure events to be in different places in spacetime: either squished closer together or stretched apart. However, events will always be in the same order (both in time and space), and events happen no matter who's looking.

You could see spacetime as a stretchy (4-dimensional!) rubber sheet, and events as dots on the sheet. You can pull at the sheet to change how far apart the dots are, but you can't swap or erase dots. Here, the pulling is analogous to changing reference frame. The behaviour of this sheet at constant velocities is described in Special Relativity.

As for the last point: they're all space-temporal! They're all derived in the sense that they differ between observers (this is called non-invariant), as I said about time above.

I hope this has cleared some things up.

• "However, events will always be in the same order (both in time and space)" That's not quite true. If 2 events are separated by a timelike interval, they retain their order for all observers, but that's not true if the separation is spacelike. Jul 20 '19 at 18:36

There is no physical meaning associated to an event. An event is a point in the spacetime manifold, where the manifold itself is a mathematical structure allowing to retrieve the effect of gravity as seen by an arbitrary observer.

To say it otherwise: any observer will see spacetime as split between space and time, which means that no observer sees an event per se. If we call physics the modelling of what we can observe, an event is not physically meaningful - it is more a mathematical tool.

In GR we describe different systems in the Universe or the Universe as a whole using four coordinates, namely three position coordinates and one time coordinate. A certain point in this four-dimensional space (pseudo-Riemannian manifold to be more precise) we call an event. Some fixed time is therefore not what we usually call an event since different for a fixed time one could have different position coordinates.

The physical meaning of an event is therefore based on the union of space and time. This was the main idea of Einstein and this union was called spacetime. An event in GR is therefore a point in spacetime whereas in Newtonian theory we had time as an event since time and position where considered differently and separately.

In GR we always deal with three spatial coordinates and one time coordinate. Of course one could regard time as a special spatial coordinate that changes the metric to a non-positive definite metric, but of course we like to give it a physical meaning.

Note: we do not always describe a system in GR with one time and three spatial coordinates. I chose to the basic approach to explain it. As described below there are exceptions to this.

• In GR we describe different systems in the Universe or the Universe as a whole using four coordinates, namely three position coordinates and one time coordinate. This is not quite accurate. For example, it's pretty common to use two null coordinates.
– user4552
Dec 2 '18 at 1:20
• Nice to know. I did not know that. I will edit the answer. Dec 2 '18 at 9:13