Metric Measurements: Explaining dx vs adx

[exmarine]

I think I have clarified one of my questions about Friedman’s metric. When an experimentalist makes a measurement of the spatial distance between two events, say along the x-axis, what exactly is his result? Is it equal to (dx), or is it (adx)?

I thought it was the former. But in cosmology, they seem to say the measured distance between galaxies must vary according to (adx). If that is true, then the time differential measurement in the Schwarzschild metric of an object, say near the event horizon, would have to approach zero, which cannot be true. You could theoretically watch such an object as long as you like, i.e., your measured time differential could be nearly infinite, but the nearly zero coefficient would cancel the time contribution to the proper interval calculation, would it not?

Thanks.

 

[PAllen]

It depends on what you measure and what coordinates you use to interpret your measure. The absence of global inertial coordinates in GR means that distance is an inherently conventional quantity, more so over large scales. The first aspect of convention is what simultaneity hypersurface you want to use - this defines a foliation. Then, you compute distance using the metric along one of your chosen hypersurfaces. Direct measurements are modeled using coordinates adapted to your chosen foliation to figure how to relate measurements to your chosen distance model. Cosmologists basically always use a foliation into hypersurfaces of constant curvature, thus manifesting isotropy and homogeneity. However, if a given observer built out coordinates trying to make them as much like Minkowski coordinates in SR as possible, they would get a completely different measure of distance. This is covered in Orodruin s insight article: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

For Schwarzschild geometry, the 'standard' coordinates are adapted to distant observers, and useless near the horizon. There is really no globally 'natural' coordinates for a BH spacetime (as there are for cosmology), but there are many choices of global coordinates that cover the whole spacetime. If you want to talk about a distance spanning the horizon, you simply must choose one of the available coordinate charts that cover the horizon. The only choice that would relate to measurements in a natural way (IMO) would be a local free fall frame (i.e. Fermi-Normal coordinates for a free fall observer crossing the horizon). This would work for a small distance away from such an observer, before, at, and after horizon crossing. So far as I know, no one has derived an explicit metric for such coordinates. Instead, they are developed as a power series of corrections to the Minkowski metric in terms of local curvature invariants.

 

[pervect]

I think I have clarified one of my questions about Friedman’s metric. When an experimentalist makes a measurement of the spatial distance between two events, say along the x-axis, what exactly is his result? Is it equal to (dx), or is it (adx)?

When the experimentalist, or the Physics forums poster, says he makes a measurement of the spatial distance along the x-axis between two events, one needs to inquire how the measurement was actually made.

I am actually imagining another context as I write this. I can't think of any experimentalist actually making a measurement of spatial distance between two events along the x-axis. But I can imagine and have seen astronomers talking about finding, through observation, "the most distant known object in the universe". It's closely related, so I will use that context instead of the original idea, as I've seen people actually talk about it.

For the "most distant object", the experimentalist measures the dopler red shift of a different object, this is usually denoted by the symbol Z, and uses that experimental measurement to calculate the number reported as distance.

The context I'm thinking of as I write this post is the context of some popular article which says "Most distant object in the universe spotted by team XXX, it is YY billion light years away". In such cases, the "YY billion" light years away" number is something that's included in the paper that "team XXX" wrote, it is not just something some journalist made up. Ideally the paper itself would explain exactly how the number was calculated, but sometimes the details are a bit lacking :(. There may or may not be some standards in the journal the paper was published that would clarify the details of the calculation. I would assume that basically the lambda-CDM model <<link>> was used, with some reasonably current (as of the time of the paper) values for the model parameters.

It might be worth pondering a simpler problem first. If one is on the Earth, and one measures the distance between a pair of survey markers <<wiki link>> , what is really being measured and how is it being measured?

The measurement, might be made by means of a theodolite <<wiki>>, or perhaps it might nowadays be made by GPS. If a PF poster was talking about measuring the distance between markers, he might mentally have in mind some different idea, perhaps he's thinking of driving a car between the two markers, and taking an odometer reading.

The point I'm trying to make is that in both of the cases I've been considering, the case of measuring the distance between the markers on Earth, and the cosmological case of the "most distant object", we gloss over a lot of details when we report the distance. We talk about the distance as if it were something we directly measure, but actually it isn't. It's the result of a process involving a mental model, where we measure other things and apply the mental model to come up with an end result, "distance".

The framework on the Earth by which we process our raw observations into "distance" might be as simple as spherical geometry, though it is possible for high precision work that a more sophisticated model is being used. The more sophisticated model would include the "figure" of the Earth, which basically arises from the Earth not being a flat plane. The details of using such a more sophisticated model might well use the concept of a metric for the surface of the Earth (it would depend on the treatment).

The framework of the astronomical measurement of the "most distant object" is General Relativity. This is also the context for the original slightly more different idea, of the "distance along the x-axis".

The end goal of the PF poster inquiring about the distance is usually trying to get a better handle on just what this framework we call "General Realtivity" is. But there's a bit of a circularity here - the details of carrying out the "distance" measurement actually invovles using the framework, the framework we are trying to understand.

I find myself being drawn to examples other than the topic of the FRW metric to talk about metrics in general. But probalby this belongs in another thread, my thoughts are drifting off-topic. If I do write something up, I'll try to put a link in this thread, though.

I think it is still on topic to say that when one seriously studies General relativity, one learns how to use the model that we call a metric to compute things that we actually measure. But when one looks closely at the things we measure, we don't generally directly measure distance at all, we measure other things and infer the distance from them.

 

[pervect]

I think I have clarified one of my questions about Friedman’s metric. When an experimentalist makes a measurement of the spatial distance between two events, say along the x-axis, what exactly is his result? Is it equal to (dx), or is it (adx)?

Let me try to take a stab at a more specific answer to this.

First, do we agree on the line element of the metric? Are you using:

$$ds^2 = -c^2 \,dt^2 + a(t) (dx^2 + dy^2 + dz^2)$$

Second, do you accept the SI definition of the meter, the distance light travels in 1 / 1/299 792 458 of a second, as the definition of what you mean by distance? Or do you have something else in mind, such as possibly one of the historical definitions of the meter? If you have somethign else in mind, what is it?

Thirdly, are you satisfied with an answer that works when the distance is sufficiently short, or are you looking for some answer that works for arbitrarily long distances?

If the answer to all three questions is yes, i.e. if you agree on the line element, you agree to use the SI definition, and you agree that you're only interested in "sufficiently short" distances, then you can use a(t)dx as your notion of distance.

Otherwise, more discussion is needed.

 

[exmarine]

Thanks for the responses. You are way ahead of me in thinking about this. I am just a beginner at studying General Relativity, and was trying to reconcile what I thought I already knew about the Schwarzschild metric with the Freidman metric.

I was thinking of relatively short distance measurements, say with meter sticks by an observer and his assistants. As he is not the proper observer, he is not at both events, so he needs some assistants. Or as you point out, maybe he could use something like a theodolite. As the Friedman is usually for astronomical distances, maybe I should move the question into the context of a distance measurement in the Schwarzschild metric? Is your answer the same there, that a short radial distance measurement would be equal to dr/(1-r_s/r) rather than just dr? Or maybe the question would be simpler to ask it about a component time interval observed. Would that be (1-r_s/r)dt or just dt?

Thanks.

 

[pervect]

Thanks for the responses. You are way ahead of me in thinking about this. I am just a beginner at studying General Relativity, and was trying to reconcile what I thought I already knew about the Schwarzschild metric with the Freidman metric.

I was thinking of relatively short distance measurements, say with meter sticks by an observer and his assistants. As he is not the proper observer, he is not at both events, so he needs some assistants. Or as you point out, maybe he could use something like a theodolite. As the Friedman is usually for astronomical distances, maybe I should move the question into the context of a distance measurement in the Schwarzschild metric? Is your answer the same there, that a short radial distance measurement would be equal to dr/(1-r_s/r) rather than just dr? Or maybe the question would be simpler to ask it about a component time interval observed. Would that be (1-r_s/r)dt or just dt?

Thanks.

You are missing a square root, but the basic idea is correct.

The short distance which we will label ds in the Schwarzschild metric with the line element you gave would be

$$ds = \frac{dr}{\sqrt{1-\frac{r_s}{r}}}$$

because
$$ds^2 = \frac{dr^2}{1-\frac{r_s}{r}}$$

One can compute this from first principles by considering in detail the path of a light beam traveling a radial path, using the fact that the Lorentz interval between emission and reception of a light pulse traveling in a straight line (aka geodesic) is zero, and the fact that a radial path is a staight line (geodesic) path. Therefore the Lorentz interval between emission and reception events for light traveling a radial path must be zero.

One needs to compute the proper time reading of a clock located at R for a light pulse to travel from R to R+$\Delta R$ and back, and then apply the SI defintion that distance = (round trip time for light ) / (2 c).

Note that the proper time reading of a clock located at R is different from the coordinate time difference, due to what's usually called "gravitational time dilation". So one might, for instance, compute the coordinate time interval for the round trip time first, then adjust it by the appropriate time dilation factor to get the proper time interval.

 

[pervect]

As far as using meter sticks goes - the basic point I want to make is that round-trip travel time of light is a perfectly fine substitute for a meter stick. Accepting this requires believing in special relativity. However, one won't get far in general relativity without accepting special relativity first.

 

[exmarine]

Ah yes, square root of course. Thanks. But I am still missing the answer to my question. It is not about how to measure distance. That is the reason I moved it to the Schwarzschild metric, and then to the time interval therein. How about I try to describe a simple “experiment”?

I make a mark on the side of my house. Then I climb a ladder and make another mark 10 seconds later – according to my wrist watch, so that is the proper time interval. You recorded the time of my first mark and remain at the foot of the ladder. Your assistant at the top of the ladder, who has synchronized his clock with yours before the experiment starts, records the time of my second mark. Then you and your assistant confer and try to calculate my proper time from the difference between your time records (dt) and your measurement of the distance between my marks with a tape measure (dr).

So my question is this: Assume the Schwarzschild metric is valid here, and the scale of the “experiment” were large enough to not be Minkowski, etc. Is that calculation (d_tau)^2 equal to (dt)^2 – (dr)^2 using your DIRECT measurements? Or must the coefficient (1-rs/r) be explicitly included and multiplied / divided times your direct measurements in the calculation? Do your direct measurements already include those coefficients? (square roots of course)

That is the question I tried to ask in the first post. “When an experimentalist makes a measurement of the spatial distance between two events, say along the x-axis, what exactly is his result? Is it equal to (dx), or is it (adx)?”

I think the first responder said the direct measurement was equal to (adx). I think that would have to mean those Schwarzschild coefficients are already in your direct measurements. The calculation would be no different than it would be for Minkowski space, which doesn’t make sense to me. The implications this has on the Friedman metric and cosmology might be significant, as the textbooks I am studying seem to use (adx) as equal to the “direct” measurements of distance. (Yes I recognize there are different “distances” in cosmology, etc.)

Thanks.

 

[Nugatory]

Your assistant at the top of the ladder, who has synchronized his clock with yours before the experiment starts...

You have a problem here: The assistant's clock cannot be synchronized with the bottom of the ladder clock, because we're assuming that the curvature effects are not negligible so the upper clock is running faster than the lower (if they were negligible, we'd be using the Minkowski metric and the problem would go away). Somehow you need to know the coordinate time, what the bottom clock reads "at the same time" that the ladder-climber is at any given height in his climb; the top clock readings are pretty much irrelevant. Note that it's not sufficient to know what the bottom clock reads when the climber reaches the top - to calculate the proper time elapsed on the climb you have to do a line integral along the climber's worldline (which is, as an aside, not a geodesic - for that he would have to jump instead of climb, thereby following a free-fall trajectory) so you need the bottom clock reading at every point.

So this version of the problem is no less complicated and confusing than your original:

When an experimentalist makes a measurement of the spatial distance between two events, say along the x-axis, what exactly is his result? Is it equal to (dx), or is it (adx)?

"The spatial distance between two events" is a tricky concept because it only makes sense if the two events happen "at the same time", and that's frame-dependent at best (consider length contraction in SR) and completely arbitrary in general (the difficulty with the clocks at the top and bottom of the ladder in curved spacetime). This limitation is implied when you say "along the x-axis", as that implies a constant t value.

However, there are special cases in which we can talk about the spatial distance between two points; we choose two spacelike-separated events, one on the worldline of each point and choose coordinates such that these events happen at the same time. For example, I might be hovering at radius $R$ in Schwarzschild coordinates, and then lower a rope until it reaches you hovering at radius $R-\Delta{r}$. We could reasonably consider the length of rope I need to be the "spatial distance" between us (although it would be a good exercise to consider exactly how we would cooperate to perform this measurement - it's trickier than it looks).

The amount of rope that I use is what it is - I can pull it back up and measure its length with my meter stick, no metric coefficients and no $\frac{1}{1-R_S/r}$ needed. However, if I want to calculate the amount of rope needed, given only the coordinates of your position and mine, then I have to use the metric coefficients and bring in that factor of $\frac{1}{1-R_S/r}$ - the length of rope will not be $\Delta{r}$, the difference between our position coordinates.

 

[pervect]

Ah yes, square root of course. Thanks. But I am still missing the answer to my question. It is not about how to measure distance. That is the reason I moved it to the Schwarzschild metric, and then to the time interval therein. How about I try to describe a simple “experiment”?

I make a mark on the side of my house. Then I climb a ladder and make another mark 10 seconds later – according to my wrist watch, so that is the proper time interval.

You recorded the time of my first mark and remain at the foot of the ladder. Your assistant at the top of the ladder, who has synchronized his clock with yours before the experiment starts, records the time of my second mark.

If I'm understanding correctly, I'm at the bottom of the ladder and my assistant is at the top of the ladder.

My assistant's clock will be running at a different rate than my clock, due to gravitational time dilation. I don't understand how you think that my assistant at the top of the ladder, and I, at the bottom of the ladder, are synchronizing our clocks.

A procedure comes to mind, but I don't think it's what you're envisioning. But let me explain the procedure that comes to my mind. It doesn't involve synchronzing proper time, but rather involves synchronizing coordinate times. At the bottom of the ladder, I have a clock that keeps proper time, and in order to get it to agree with the coordinate time t used in the Schwarzschild metric, I somehow rate adjust my clock. This could be built into the clock, probably some sort of frequency divider. Since I also have an interest in proper time for this problem, though, I'd rather imagine that my clock has two digital time readouts. One time readout keeps coordinate time, and the other time readout, keeps proper time. I label the two readouts so they don't get confused.

My assistant, at the top of the ladder, has the same sort of clock, with two readouts. His time adjustment factor between coordinate time and proper time is different than mine, however, as he is at the top of the ladder and I'm at the bottom, so the gravitational time dilation factors are different. So the frequency divider setting needed to turn proper time into coordinate time would be different for him than me.

While my assistant and I can't keep the proper times of our clocks synchronized as they run at different rates, we can keep our coordinate times synchronized.

Given this setup, though, I don't quite see what the person climbing the ladder is doing, precisely. My basic guess, however, would be that you didn't realize that two clocks keeping proper time at different altitudes would run at different rates, and that you'd want to revise your scenario slightly.

Skipping ahead to your end goal, though, if you let $dp = \sqrt{1-r_s/r} \,dt$ and $dq = dr/\sqrt{1-r_s/r}$, then we can write the Schwarzschild metric as -dp^2 + dq^2 (ignoring the theta and phi coordinates for now).

Then p and q are local coordinates (that are not Schwarzschild coordinates,but a transform of them) so that $d\tau^2 = -dp^2 + dq^2$ with the sign convention I favor. I think you suggested the opposite sign convention, which is valid, I just have a way of doing things that I'm used to and I stick to it, as I make less errors that way. I haven't specified p as a function of t directly, to do that you'd need to integrate the differential constraint.

So we can create coordinates that do what you want (not everywhere, but in the region around some particular observer). And they are useful. They're just not the same as Schwarzschild coordinates. We can only pull off this trick in some small region, however. We can't make $d\tau^2 = -dp^2 + dq^2$ true everywhere, but we can make it true in a local region around some specifically chosen point.

Near the chosen point, these adjusted coordinates are Minkowskii, and we can tie the p coordinate to proper time of an observer at that point, and tie the q coordinate to proper distance in the radial direction.

 

[exmarine]

You guys are right! I forgot all about the clocks at different elevations running at different rates. Now I need to study the rest of your comments to see if I can understand them.
Thanks.