Modeling of measurements in relativity

[Ken G]

There is no other way to express physics adequately than mathematics.

I would say there is no way to predict experimental outcomes than mathematics. But "expressing physics" can be rather different. For example, how many times have we asked ourselves, "I see that the mathematics works out this way, but I want a physical explanation." Indeed, I have found that asking myself for a physically motivated explanation has often led to key insights. If one says "there's nothing but the mathematics," then special relativity reduces to "the Lorentz transformation with the Einstein simultaneity convention," and that's it, done-- all the invariants follow, all the experimental outcomes are extended to all SR frames. So we would never need to mention "length contraction," or "time dilation", or the "relativity of simultaneity", for none of those things are necessary if we have the Lorentz transformation. But we do mention those things-- because they give us a sense of something physical that is going on. It can be problematic, to be sure-- but we seek the physical explanation all the same.

If you have an idea to explain this better, let me know.

You simply drop the uniqueness requirement-- you don't imagine that you are looking for "the" physical explanation, only "a" physical explanation. So you say "let's look at the situation from the point of view of observer X, how do they account for what is going on?", and we look for a physical language that translates the mathematics into more everyday terms, from that perspective. If the observer is in the initial rocket frame, we say the rope length contracts and can no longer span the constant distance between rockets. If the observer is on the rope, we say the lead rocket burns fuel faster and accelerates more. If the observer is in the eventual inertial frame of the coasting rockets, we say the lead rocket took off first. Each of these explanations sounds like everyday language, so they give us a sense of physical understand that goes beyond "the mathematics works out that way."

Further, with science we don't answer "why questions", because by definition science is an as precise as possible description of nature based on objectively observable and quantifiable phenomena, no more nor less. It does not answer, "why" the physical laws we discover from precise measurements and theory building are as they are.

I would say we do answer "why" questions all the time (why is the sky blue, why does the rope break, why does light bend in a lens, etc.), but we do not answer them uniquely. It's what we mean by an explanation. The mathematics of a chosen theory is unique, but the explanations are not.

 

[Ken G]

You can answer the question without having to pick a reference frame or coordinate system at all. See below.

The mathematics of relativity never requires coordinates, it is a coordinate-dependent theory, so it's not surprising you can seek a coordinate-free answer (and more power to you, it is a powerful way to frame relativity theory). But physics calls for (at least) two things more than just the mathematics. It calls for observations that can check the mathematics, and it calls for physical explanations. The observations are essential, and we do not always have access to invariants-- we often have to interpret our observations from the perspective of a given coordinate system. Hence we cannot do physics in a purely coordinate-free way. The physical explanations are not essential, if our only goal is to predict observations. That's the "shut up and calculate" school. But in all honesty, I've never actually seen anyone who only shuts up and calculates, even those who claim they only do. They always seek something more.

The simplest mathematics that expresses, in a way independent of frames and coordinates, your statement in ordinary language that "the proper distance between the ships increases with time" is the expansion scalar of the congruence of worldlines describing the ships and the string. This expansion scalar is an invariant, and it is positive; that is the invariant way of saying that the proper distance increases.

Yet this statement more or less proves my point. I do not dispute the extreme elegance and power of seeing the solution in these terms. But we must also recognize that physics has other goals than just to find the most elegant path to an answer-- it also seeks to be a good communicator of ideas to the public. This is one of the (many) reasons why we also seek what we can call a "physical explanation", and use words like "length contraction", which is something that the public can kind of get their heads around (even though it is rarely the most elegant path to a solution). However, what the public does not always realize is that the "physical explanations" they are getting are not unique descriptions of what is "really happening", they are just a kind of picture coming from a certain perspective. (The purely mathematical and coordinate-free answer seems closer to what is "really happening", but it also cannot be that, because it requires adopting some particular theory that we cannot know is exactly correct.)

 

[PAllen]

... The observations are essential, and we do not always have access to invariants-- we often have to interpret our observations from the perspective of a given coordinate system. Hence we cannot do physics in a purely coordinate-free way...

On this specific point, divorced from philosophy, I have to disagree. The only way a measurement made by some device is not an invariant is if two observers can disagree on what a given instrument reads following a given measurement. This is absurd. Thus, any measurement actually made is an invariant, and in relativity, that specifically means any measurement must be definable in terms of scalar contractions or their integrals.

 

[Ken G]

On this specific point, divorced from philosophy, I have to disagree. The only way a measurement made by some device is not an invariant is if two observers can disagree on what a given instrument reads following a given measurement. This is absurd. Thus, any measurement actually made is an invariant, and in relativity, that specifically means any measurement must be definable in terms of scalar contractions or their integrals.

What I'm saying is, how would you measure the proper distance between two events, let's say a supernova explosion, and your act of detecting it? That's an invariant, but you don't measure it. So to test that relativity is getting that invariant right, you are going to work in some coordinates. Are you not?

 

[PAllen]

What I'm saying is, how would you measure the proper distance between two events, let's say a supernova explosion, and your act of detecting it? That's an invariant, but you don't measure it. So to test that relativity is getting that invariant right, you are going to work in some coordinates. Are you not?

I agree that is an invariant that you can't measure directly, because you can't materialize is specific hypersurface (which is needed to even define a proper distance) over large distances. However, to predict a set of invariants you can measure (e.g. red shift vs. luminosity distribution) you don't need to even worry about getting that right. That unobservable proper distance is just part of mathematical framework for computing the invariants you can measure. In practice, you need some coordinates for that, but it doesn't matter which coordinates you use.

 

[PeterDonis]

we do not always have access to invariants

Yes, we do. In fact, they're the only things we have direct access to. See below.

what the public does not always realize is that the "physical explanations" they are getting are not unique descriptions of what is "really happening", they are just a kind of picture coming from a certain perspective.

I disagree. Explaining observables in relativity does not require choosing a "perspective". It just requires correctly modeling the observing apparatus with geometric objects, the same way you do the objects being observed. The "relative velocity" of two objects, for example, is just the inner product of their 4-velocity vectors at the event where their worldlines cross. That's an invariant. Similar invariants can be constructed for every observable; you just pick the appropriate 4-vectors, tensors, whatever, and form scalars out of them by contracting so that there are no free indexes left.

In fact, even these 4-vectors, tensors, etc. are constructed from invariants. The "components" of a given 4-vector are really just contractions of that 4-vector with a particular set of 4 "basis" vectors--i.e., they're a set of 4 invariants.

What you may be referring to by a "perspective" is that different observers might think different invariants are important. For example, if observers A and B are moving on different worldlines, and they both want to measure the "speed" of an object passing by them, A is going to be interested in the contraction of his 4-velocity with the object's, at the event where the object crosses his worldline; but B is going to be interested in the contraction of his 4-velocity with the object's, at the event where the object crosses his worldline. These are different invariants, so of course they can have different values. But if you ask B what velocity A will measure, or vice versa, they will both give the same answers; i.e., once you've specified a particular invariant, all observers will agree on its value.

So when you say physical explanations are not unique descriptions of what is "really happening", I think you are misstating it. What should be said is that different "perspectives" are descriptions of different invariants. The difference in "perspective" is entirely a question of which particular invariants you choose to focus on; it's not a difference in "what is really happening", because all the invariants have the values they have regardless of which ones you pick to emphasize.

 

[Ken G]

I agree that is an invariant that you can't measure directly, because you can't materialize is specific hypersurface (which is needed to even define a proper distance) over large distances. However, to predict a set of invariants you can measure (e.g. red shift vs. luminosity distribution) you don't need to even worry about getting that right. That unobservable proper distance is just part of mathematical framework for computing the invariants you can measure. In practice, you need some coordinates for that, but it doesn't matter which coordinates you use.

It never matters what coordinates you use. You confirmed my point: you need some coordinates. So you cannot tell that relativity is giving you a consistent picture of the things that are invariants, without using coordinates. This is very much analogous to giving physical explanations of phenomena-- they often involve coordinates also. The point is, physics tells stories, and to do so, it uses coordinates. You can use any coordinates you want-- but the stories you tell will sound different.

 

[Ken G]

Yes, we do. In fact, they're the only things we have direct access to. See below.

Yet I asked a specific question. We talk about the proper distance between a supernova explosion and the event of our detecting it. To check relativity, we need to do observations to check whether that invariant indeed turns out to be an invariant. This requires observations that are not that invariant, so we need to translate the observations we have into coordinates to check the invariant.

Let me take a more concrete example-- the lifetime of a relativistically moving particle in the laboratory. We know there is an invariant proper time for that particle, but we don't have a clock on the particle, we have two clocks in the laboratory, at opposite ends of the motion of the particle over its lifetime. We must use those two clock measurements, which are synchronized in a way that reflects a given coordinate system, to determine the proper time for the elementary particle. That's a classic example of what I'm talking about-- to talk about the "time dilation" that allows the particle to "seem to exist" for a longer time than its own proper lifetime, we need to invoke a coordinate system. But even more than that-- we also invoke those coordinates to test that the proper lifetime of the particle is indeed what we expect. The theory of relativity says the lifetime is an invariant, but tests of relativity don't say that, they test that. It doesn't matter what coordinates we choose, but we have to do the experiment, and that means we need a language for talking about what happened, and that means we need to choose coordinates. This is what I mean by how choices of coordinates are not unique, but they do give rise to a language for talking about what happened-- a language for giving physical explanations. We do this all the time-- indeed any time the term "time dilation" appears in a physics book (which is a lot!).

I disagree. Explaining observables in relativity does not require choosing a "perspective".

I guess a lot hinges on the word "requires" here. No one can disagree that if we look at explanations of observables found in various places, be they course websites or physics books, we will find lots of examples of coordinates being chosen. We will find statements like "gravity makes things fall downward", and "the Earth orbits the Sun", both of which are of course examples of choosing coordinates to produce a language suitable for making physical explanations. So you must be talking about whether or not language like this is required. So I will first point out that it is often used, and this is an important thing to recognize.

The reason I said I thought it was required is that all observations are indeed done from some perspective-- the observer has a frame of reference. The observation itself is only an invariant if we take account of this reference frame-- the invariant is only that "observer in frame F will observe X." If you drop the first part, i.e. the "perspective", you lose the invariant.

So when you say physical explanations are not unique descriptions of what is "really happening", I think you are misstating it. What should be said is that different "perspectives" are descriptions of different invariants.

I don't think those statements are saying different things. Let me give an example. A star is moving away from us, and we measure a redshift. We ask, "why is there a redshift"? One possible explanation is "there is a relative velocity between the star and us." But of course that is not a coordinate independent explanation, because in an accelerating reference frame, the distance between us and that star could be decreasing, and we could not then say that there is such a relative velocity. In that frame, the redshift would be due to a different explanation. So the invariant here is only that we observe a redshift-- the why we observe a redshift is coordinate dependent. The explanation will be framed in terms of invariants either way, but different invariants will be in the story, because of the different language, because of the different coordinates.

The difference in "perspective" is entirely a question of which particular invariants you choose to focus on; it's not a difference in "what is really happening", because all the invariants have the values they have regardless of which ones you pick to emphasize.

That's my point, it's not a difference in what is happening, it is a difference in the language we choose to say what is happening.

 

[PAllen]

It never matters what coordinates you use. You confirmed my point: you need some coordinates. So you cannot tell that relativity is giving you a consistent picture of the things that are invariants, without using coordinates. This is very much analogous to giving physical explanations of phenomena-- they often involve coordinates also. The point is, physics tells stories, and to do so, it uses coordinates. You can use any coordinates you want-- but the stories you tell will sound different.

OK, I agree that with different coordinates you tell different stories for the same observations. Peter (above) explains how even these different stories may instead be considered different choices of invariants to emphasize. However, I have no problem with the idea that different coordinates tell different stories for the same observations. I also agree that no on, in practice, computes observables without using coordinates. Yet, ignoring calculation pragmatics, I agree with Peter's point that you don't need any coordinates to tell different stories.

I'll give a concrete example of this. Using parallel transport of different choices 4-vectors on a null geodesic, you can tell the following different stories:

- All spectral shifts in both Special and General Relativity (including gravitational and cosmological) are just Doppler shifts.
- All spectral shifts in both Special and General Relativity are gravitational (including pure SR Doppler and cosmological)

 

[PAllen]

Yet I asked a specific question. We talk about the proper distance between a supernova explosion and the event of our detecting it. To check relativity, we need to do observations to check whether that invariant indeed turns out to be an invariant. This requires observations that are not that invariant, so we need to translate the observations we have into coordinates to check the invariant.

No we don't. And we can't. We never need to check that unobservable invariant. We only delude ourselves if we think we have. All observables can be computed without even using that unobservable invariant, so how can we in any sense verify it? The most we can say is that observations are consistent with calculations done in coordinates that use that invariant (but are also consistent with no use of that invariant).

 

[PAllen]

The reason I said I thought it was required is that all observations are indeed done from some perspective-- the observer has a frame of reference. The observation itself is only an invariant if we take account of this reference frame-- the invariant is only that "observer in frame F will observe X." If you drop the first part, i.e. the "perspective", you lose the invariant.

I disagree. You need not associate a frame to an observer, and doing so has nothing to do with measurements being invariant. I can posit 10 observers, none of whom are comoving with an instrument. All will agree on whether the instrument reads 5 versus 10, irrespective of wither we do any analysis in an frame in which any of the observers or the instrument are at rest.

 

[Dale]

Measurements are invariant.

Thread closed.