# Units for Coordinates?

1. Sep 8, 2014

### pervect

Staff Emeritus
I was reading "Time scales in the context of general relativity" Bernard Guinot, and a few other papers whose names I forget, and was surprised that there was apparently some desire by some physicists to give coordinates units.

It seems that the current recommended practice is that coordinates don't have units (which is the position I take), but I fail to understand why this is not unanimously accepted, nor why one would want to assign units to coordinates, such as the above mentioned "TCB seconds".

A change in the T coordinate depends on which coordinate system you use, and not to anything actually measurable. Not being a physical quantity in and of itself, it seems to me to be wrong and confusing to assign a unit to a change in coordinate.

2. Sep 8, 2014

### stevendaryl

Staff Emeritus
It seems to me that coordinates can have any units you like. The usual $x,y,z$ have units "cm" (or whatever). The only thing that is physically meaningful is $g_{\mu \nu} dx^\mu dx^\nu$, which must have dimensions length2. So you can freely move units back and forth between $g_{\mu \nu}$ and $x^\mu$.

But arbitrary coordinates can't really have units. It seems that it's only special coordinates that it makes any sense to give units to.

3. Sep 8, 2014

### Staff: Mentor

I am with you on this. I prefer dimensionless coordinates. That lets you put all of the dimensions in the metric.

4. Sep 8, 2014

### ghwellsjr

Is this just a GR thing or does it also apply to SR?

What do TCB-second and TCG-second mean?

If we have dimensionless and unit-less coordinates, how do we apply the Lorentz Transformation to events?

Are we supposed to have a spacetime diagram without grids?

5. Sep 8, 2014

### ghwellsjr

Another question:

What relativistic effect is the problem here?

6. Sep 8, 2014

### Staff: Mentor

Is this really current practice? Do physicists normally write papers with time dimensionless, for example? (Which would mean a given time interval would be the number of Planck intervals.)

7. Sep 8, 2014

### pervect

Staff Emeritus
It would mainly apply to GR, or at least situations where your metric coefficients are not unity. If your metric coefficients are unity, then a unit change in coordinate is equivalent to a unit change in distance, making the urgency of the need for distinction low, until such time as one re-introduces non-unity metric coefficients.

In practice, though, the metric coefficients for measurements in and around a gravitational field are not and never have been unity, it's just that the corrections are so small that they can usually be ignored. With the advance of precision in measuring techniques, there are more and more situations where the corrections cannot be ignored. People who have in the past conflated coordinate changes with distances can experience immense and difficult-to-resolve confusion in this circumstances.

They are references to the time unit in the ICRS (International Celestrial Reference System) and the ITRS (International Terrestrial Reference System), two coordinate systems recommended by the International Astronomical union. The ICRS uses a barycentric coordinate system whose origin is at the solar system barycenter. See the wiki article on TCB time for an overview, http://en.wikipedia.org/wiki/Barycentric_Coordinate_Time. The ITRS uses a geocentric coordinate system whose origin is at the Earth's center. See the wiki article on TCG time for an overview. http://en.wikipedia.org/wiki/Geocentric_Coordinate_Time

One of the major sources of discontent is, I think, the fact that TCB and TCG time don't tick at the same rate as clocks on the surface of the Earth (i.e. the geoid) do, making them have a scaling factor from the commonly used atomic time scale (TAI time).

I would say that you'd still draw and use grids, you'd just have to recognize that in curved space-time the length of the sides of the grid are not constant. It's rather similar to the issue that arises when one draws grids on the surface of the Earth. One might well plot the course of a ship sailing on the ocean on a navigational chart with lattitude and longitude drawn as a square grid. When interpreting this chart, though, one would need to recognize that the planar maps of the Earth are inherently distorted, and that a degree of longitude, though it appears square on your navigational map, does not represent a constant distance. Instead, a degree of longitude represents a distance that changes with lattitude, described by the metric of the surface of the Earth, and the "square" grids as drawan on the map, when drawn on the surface of the Earth, are no longer necessarily square in terms of the distances as measured on the surface of the Earth.

8. Sep 8, 2014

### pervect

Staff Emeritus
The IAU is recommending NOT to introduce a plethora of "coordinate seconds" and/or"coordinate meters", feeling (rightly, IMO) that this will cause confusion, with people believing that seconds and/or meters are different when you move from place to place because the coordinates are different. (Reminds me of PF a lot, actually, we see a lot of that here).

There isn't any push to go to Planck intervals, though.

[add]I think it's OK to assign units to ds^2 when you define or describe what your coordinate system is, at least as long as you keep all those annoying factors of "c" so that you are able to do that. But you can assign units to ds^2 without assigning units to the coordinates themselves.

Current practice isn't necessarily following the IAU recommendations, so it's not clear what "current practice" is. There is at least some discussion in papers on what the best practice should be.

Last edited: Sep 8, 2014
9. Sep 8, 2014

### Staff: Mentor

You can, in principle, yes, but this is definitely *not* common practice. Most of the literature that I've seen uses dimensionless metric coefficients for the most part, the only common exception being angular coordinates. Consider the standard way of writing the Schwarzschild line element, for example.

(One case where dimensionless coordinates *are* often used is FRW spacetime, putting all the dimensions in the scale factor. Even in this case, however, the time coordinate still has dimensions.)