# How can we define a frame of reference in general relativity?

I have started reading general relativity. (A First Course in General Relativity, Bernard Schutz).

I am finding very hard to understand a frame of reference. When I was reading special relativity (well,still reading) I pictured a frame of reference being grid with infinite number of clocks everywhere in space and where all clocks are synchronized. When an event takes place the clocks record the time (the event takes place). But in general relativity clocks run at different rate in different place so how can we talk about synchronized clocks? So how can we record events?

Suppose the time is recorded by a person (or a device) at the origin ,would coordinate time interval be time interval measured between two events by the clock of the person?

• we usually use Riemann (or Gaussian, they mean the same) normal coordinates for the sake of simplicity. It is merely an infinitesimal coordinate system on a curved manifold such that it is approximately flat, and for that coordinate system, $g_{\mu\nu} = \eta_{\mu\nu}$. For example, on a curved surface, you can select any area sufficiently small so that it is approximately flat, and you carry out your operations (well, mostly) in those coordinate systems in general relativity. – GRrocks Apr 25 '15 at 7:08

In general relativity, it just has to be continuously differentiable. If you walk along a grid line, it can't suddenly turn, and the clocks can't suddenly change speed. Beyond that, pretty much anything goes. You don't even have to cover everything with a single map, but you do have to have extra maps to make sure it all gets covered somewhere.

For example, if you wanted to look at the surface of a sphere, you could use the longitude and latitude for your grid, along with some extra maps for the north and south poles which don't get covered right. In general relativity you have to deal with time as well as space, but it works pretty much like that.

reading special relativity [...] I pictured a frame of reference being grid

Of course there is no definitive requirement for the grid constituents to be rigid with respect to each other, or being in any particular way "regularly spaced" or "regularly moving". Required is (only)

• for the grid constituents to be distinctive,

• for any two grid constituents never to meet (otherwise the grid would have to be called "degenerate"), and

• arguably, without any "edges" or "horizons" or "gaps".

To summarize: within a flat region of spacetime, a reference frame may be an inertial reference frame (as explained by Rindler: "simply an infinite set of point particles sitting still in space relative to each other"), but it doesn't have to.

with infinite number of clocks everywhere in space and where all clocks are synchronized. When an event takes place the clocks record the time (the event takes place).

In order to "record an event" surely it is not necessary to assign to it some particular $t$ coordinate value, by means of (or representative of) some particular clock. It should suffice to note

• who (which distinct constituent of "the grid") took part in the event under consideration,

• that this participant also records which observations (especially of other grid constituents, having taken part in other events) she had made in coincidence with having taken part in the event under consideration,

• and, arguably, that each participant may judge and record the sequence of events in which they took part.

This general, explicitly coordinate-free notion of "reference frame" is readily applicable to curved regions of spacetime; however, with "edges" or "horizons" or "gaps" unavoidable in certain cases. It's also identified as any applicable "timelike congruence".

There are different kinds of frames. A common frame to use is a coordinate frame. For that all you need to imagine is each region of spacetime has a coordinate system that you can use in that region to find and label all the events in that region.

An advantage to this is that you can practice using arbitrary coordinate systems even while still doing Special Relativity, just do Special Relativity in polar coordinates, then in cylindrical coordinates, then in spherical coordinates, etcetera. Learn how to use all these different coordinates system, but don't (yet) worry about curvature or General Relativity. Same things with tensors, you can learn to use them in Special Relativity.

Your textbook does these things actually. So maybe your confusion is that you think the General Relativity section has started before it has actually started, or maybe you skipped ahead.

Now, however, there is a completely different way people use the word frame, which is to refer to a frame of vectors. In that case each event has some vectors, enough to specify all the velocities and such at that point. This is very different, because these vectors give you directions but they do not give you coordinates. In a Euclidean space they might look like the exact same thing (in a Euclidean space, vectors and covectors also look the same, but they are different), so let's tell the difference.

I will set up a polar coordinate system to explain the difference. In polar coordinates you can move outwards one radial coordinate (so one meter out), then over one angular coordinate (so one radian over), then back in one radial coordinate system (so one meter in) then back over one angular coordinate (so one radian back). You end up back exactly where you started. Coordinates locally look like distorted graph paper, they always have that property.

But what if you had a frame of vectors that everywhere had a vector pointing radially outwards and another one pointed in the clockwise direction? If you moved one meter outwards, then one meter counterclockwise, then one meter radially inwards, then one meter clockwise then ended up moving clockwise. So a coordinate system and a frame of vectors are different.

Coordinate systems are something you'll have to learn to deal with, and how to do calculus in them. And even in Special Relativity, you can have a metric that varies from place to place in your coordinate system. The polar, cylindrical, and spherical coordinate systems are examples of that.

Later on, you'll also have to deal with curved coordinate systems, which means the metric must vary from place to place, but it will do so in particular in a way that affects parallel transport. Thus geodesics can converge, which will be how massive bodies exert tidal forces.

You can't measure distances between events with a coordinate system alone. But if you already have a coordinate system can do it if you also have a metric. But if you want to move around you can completely describe your motions by saying how the coordinates change. It helps to be an expert in describing changes through changes in coordinates, because then you can describe the metric in terms of how actual metric distances are related to coordinate distances. So you can use coordinate distances as a foundation to build upon. And again, doing it for interesting coordinate systems in the context of regular euclidean geometry or in regular Special Relativity is a good practice and you should consider it and your textbook does it, so your textbook is assuming you have that facility.

So now let's get to a frame of reference. A frame of reference is definitely a coordinate system. However if you want to bring in clocks and rulers they measure actual distances, so they will have a nontrivial relationship to your coordinates, and the metric will tell you exactly how they are related.

Maybe by frame of reference you mean a coordinate system with a metric. Having them both. An inertial reference frame is a special frame, one where the coordinates represent the motions of test particles subject to no external forces. These strictly only exist locally.

• Actually I was trying learn relativity, first by asking questions or also as you said I skipped a little. thanks for the ans, it was helpfull. – Paul May 6 '15 at 20:52
• Timaeus: "If you want to measure distances between events" ... or rather: intervals $s^2$ between events, or related quantities such as $\text{sgn}[~s^2~]~\sqrt{s^2~\text{sgn}[~s^2~]}$ ... "you'll need the coordinate system and the metric [tensor]" -- No: we certainly don't need coordinates for measuring geometric relations. Instead, we may (as an afterthought) derive the metric tensor from measured intervals, for any suitable (differentiable) assignment of coordinates to events. – user12262 May 6 '15 at 21:56
• @user12262 Thank you, I've edited my answer. – Timaeus May 6 '15 at 22:47
• Timaeus: "Thank you, I've edited my answer." -- Thank you, in turn. You've substituted the phrase: "You can't measure distances between events with a coordinate system alone. But [...]" -- Two remarks: 1: It's still incorrect and puzzling that you refer to "distances between events". If you like to avoid mentioning the measurement of intervals between events, or of related quantities, and you like to mention the measurement of distances, then you should refer to "measurement of distances (or more generally: separations) between participants (as grid constituents), or of durations". – user12262 May 7 '15 at 5:15
• 2: You seem to suggest that if you're given a coordinate system then you're not (necessarily, by definition) already given all geometric relations (such as distances, or durations, or intervals); so there was indeed occasion and need for measuring geometric relations in addition to what's given already. How, then, would you distinguish a coordinate system from any plain labelling with n-tuples of real numbers?? (See also "What is the most general definition of a coordinate system?" (PSE/q/178907).) – user12262 May 7 '15 at 5:15