Recommendaton for Clarifying Special Relativity

[PeterDonis]

How would you express that constraint (Lorentz covariance) without at least implicitly referring to Lorentz transformations and the operationally defined systems of coordinates that they relate?

By the fact that all actual observables can be expressed as Lorentz scalars, which can be written as coordinate-free tensor expressions with no free indexes. The operationally defined measuring apparatus can be modeled as a 4-tuple of orthonormal vectors, also expressed as coordinate-free tensor quantities; observables are then simply contractions of other coordinate-free tensor expressions with the expressions describing the appropriate members of the 4-tuple.

To put the question differently, how would you state the physical principle(s) underlying special relativity?

See above.

Often in introductory presentations of special relativity the word "observer" is used incessantly, and this leads to all kinds of confusion

Yes, "observer" is one of those abused words and I probably shouldn't have used it to describe what I was thinking of. Hopefully the above makes it clearer.

 

[PeterDonis]

I think what Peter meant is that people often confuse coordinate systems with Lorentz frames.

That's a big part of what I was thinking of, yes. But it's also important to realize that "Lorentz frames" can be defined in a coordinate-free manner. Coordinates are a calculational convenience, not a necessary part of the formulation of the theory.

 

[WannabeNewton]

Certainly, I agree. This ties into what you said immediately above, about the Lorentz frame operationally defining the measuring apparatus.

 

[Samshorn]

By the fact that all actual observables can be expressed as Lorentz scalars, which can be written as coordinate-free tensor expressions with no free indexes.

Up to that point you're just re-stating the fact the physical phenomena are Lorentz covariant, without giving any physical meaning to the assertion. Now, my question was "How would you express Lorentz covariance [in a physically meaningful way] without at least implicitly referring to Lorentz transformations and the operationally defined systems of coordinates that they relate?" Your answer is apparently here:

The operationally defined measuring apparatus can be modeled as a 4-tuple of orthonormal vectors...

Well, it isn't just the measuring apparatus that must have operational meaning. The thing being measured must have operational meaning too. Also, this "orthonormal 4-tuple" implicitly invokes inertia to establish orthonormality (otherwise it would have no physical meaning). Also, this all amounts to a local inertial coordinate system, so up to this point it would seem that we're in agreement that you can't really express Lorentz covariance of physical phenomena in an operationally meaningful way without referring to an operationally defined system of measurement. However, you actually disagree with this, and your rationale is apparently contained in the following words, where you say these 4-tuples are:

...also expressed as coordinate-free tensor quantities; observables are then simply contractions of other coordinate-free tensor expressions with the expressions describing the appropriate members of the 4-tuple.

But none of those tensor expressions have any physical meaning unless you define, in an operational way, what the terms mean. It makes no sense to claim that only the contractions of your tensor expressions have physical meaning, and not the components. Of course, the numerical values of the components depend on the choice of coordinate systems, but this does not render them meaningless - quite the contrary, it is what makes them physically meaningful. They would be meaningless only if our choices of coordinate systems were meaningless - which they better not be, or we have no physical theory at all.

Look, even when posing a question (let alone answering one) we need physically meaningful coordinates. For example, if someone says "Suppose I'm moving directly away from you at speed v. What is the Doppler shift?" Well, that question doesn't even have any meaning unless you can physically interpret what it means to have a speed v. Presumably the questioner means the distance per time for an inertial coordinate system. According to your thesis, we can't even speak about speeds, spatial lengths, kinetic energy, coordinate time intervals, etc., any quantity which is coordinate-dependent, which makes your theory fairly useless. Being coordinate dependent is not the same as being meaningless - quite the contrary.

I think it would help if you thought about how you would answer this question: How would you test Lorentz invariance?

 

[PeterDonis]

The thing being measured must have operational meaning too.

Sure; it's a worldline or world-tube, a geometric object, and operationally measurements of it are defined by picking out events where the worldline or world-tube is intersected by other worldlines or world-tubes corresponding to the measuring apparatus.

Also, this "orthonormal 4-tuple" implicitly invokes inertia to establish orthonormality

Meaning, it invokes the metric? Yes, but the metric itself is a coordinate-free geometric object.

Also, this all amounts to a local inertial coordinate system

No, it doesn't, because you don't have to define coordinates to do any of what I said. The fact that one usually does so is, as I said, a calculational convenience, not a necessity.

For an example of a physical analysis that doesn't define coordinates at all, see MTW's "centrifuge and photon" example; I don't have my copy handy, but I think it's in the first or second chapter. The entire analysis is done without ever defining coordinates at all, purely by considering coordinate-free geometric objects representing the photon and the centrifuge.

up to this point it would seem that we're in agreement that you can't really express Lorentz covariance of physical phenomena in an operationally meaningful way without referring to an operationally defined system of measurement.

I agree as long as "operationally defined system of measurement" does not include coordinates, since they're not necessary.

It makes no sense to claim that only the contractions of your tensor expressions have physical meaning, and not the components.

Huh? That's exactly what is normally done! Tensor components are coordinate-dependent and have no physical meaning; only contractions, with no free indexes, can correspond to direct observables.

Furthermore, we can define tensor components in terms of contractions, rather than the other way around; each component is the contraction of the tensor with a particular pair of vectors in our 4-tuple. MTW goes into this in some detail.

Of course, the numerical values of the components depend on the choice of coordinate systems, but this does not render them meaningless - quite the contrary, it is what makes them physically meaningful. They would be meaningless only if our choices of coordinate systems were meaningless - which they better not be, or we have no physical theory at all.

Huh? There is nothing forcing us to choose physically meaningful coordinates--indeed, as I said before, there's nothing forcing us to choose coordinates at all. The fact that we usually choose coordinates where at least some of them have an easy physical interpretation is, as I said before, a calculational convenience, not a necessity.

Look, even when posing a question (let alone answering one) we need physically meaningful coordinates. For example, if someone says "Suppose I'm moving directly away from you at speed v. What is the Doppler shift?" Well, that question doesn't even have any meaning unless you can physically interpret what it means to have a speed v.

You posed the question backwards. The Doppler shift is the direct observable; the relative velocity v is a physical interpretation of what the observable means.

Presumably the questioner means the distance per time for an inertial coordinate system.

Yes, he might. So what? He's not trying to construct a theory from first principles; he's just asking a question that his intuition leads him to ask. As a matter of convenience, once again, we don't insist on always building up questions from first principles. But if we're going to talk about foundations, we can't make arguments based on convenience.

According to your thesis, we can't even speak about speeds, spatial lengths, kinetic energy, coordinate time intervals, etc., any quantity which is coordinate-dependent

Sure we can; we just have to define them in terms of coordinate-free expressions, direct observables, instead of the other way around. For example:

We define relative speed in terms of observed Doppler shift.

We define spatial lengths in terms of contractions of the spatial vectors in our 4-tuple with the spatial vectors describing the object whose length we are interested in.

We define energy as the contraction of our 4-velocity (the timelike vector of our 4-tuple) with the 4-momentum of the object whose energy we are interested in; kinetic energy is then just this energy minus the invariant length of the object's 4-momentum.

Coordinate time intervals could be defined several ways; the easiest one I can think of is to use observed Doppler shift and proper time along the observer's worldline, but there are others.

And so on.

I think it would help if you thought about how you would answer this question: How would you test Lorentz invariance?

The same way it's already been tested. I don't think we disagree at all about experimental results; we're talking about the logical structure of the theory we use to describe them.

 

[WannabeNewton]

What Samshorn is describing with regards to measuring the components of tensor fields in an orthonormal frame field and the relation of the measurement of these components to special covariance (i.e. what the components look like in another orthonormal frame field under the transformations coming from the isometry group of the space-time) is what Wald goes into detail on in the first section of chapter 13, for anyone interested.

 

[Samshorn]

Quote by Samshorn: "Also, this "orthonormal 4-tuple" implicitly invokes inertia to establish orthonormality."

Meaning, it invokes the metric?

No, meaning that orthonormality is based on the isotropy of inertia (just as with an inertial coordinate system). This is (for example) ultimately how we decide what time direction is orthonormal to the space directions for any given state of motion. I'd say this is the core of our disagreement. You want to invoke things like orthonormal 4-tuples without acknowledging that it is tantamount to an inertial coordinate system, and that orthonormality has non-trivial physical significance, based on empirical phenomena that could (a priori) have been otherwise.

Sure; it's a worldline or world-tube, a geometric object...

It's only a geometric object if we have a metric, and the metric must be operationally defined in order to have physical meaning. Now, you say the metric is a geometric object too, but again that is only true if we have a metric, and the only way to break this circularity is to confess that there must actually be operationally recognizable coordinate systems and corresponding metrics with specific components - and that this is the foundation of the theory. Physics is not tautological.

We can't do anything (much) with just the generic idea of a metric. To make actual predictions, we need a definite metric, and although it can be expressed in terms of different coordinate systems, it obviously isn't arbitrary or physically meaningless. We need some way of translating observable facts so as to constrain and determine this metric sufficiently to answer questions. Ultimately we can only do this by observing the behavior of phenomena, and establishing the coordinated measures that collectively amount to a coordinate system (although of course it need not be expressed formally as a coordinate system).

No, it doesn't, because you don't have to define coordinates to do any of what I said. The fact that one usually does so is, as I said, a calculational convenience, not a necessity. For an example of a physical analysis that doesn't define coordinates at all, see MTW's "centrifuge and photon" example...

Ah yes, MTW, the apostles of the coordinate-free faith. That example (paragraph 2.8) is a classic for how it tacitly smuggles in all the information, not least when they blithly equate the wavelength ratio with the energy ratio, and when they claim the magnitude of the "ordinary velocity" of the rim is unchanging - which of course is only true if the centrifuge is at rest in an inertial coordinate system - and only if we understand what "ordinary velocity" means, which they aren't even entitled to talk about in the absence of coordinates, etc. That example just illustrates my point. People who think they are dispensing with inertial coordinates never really are. The same applies in Newtonian mechanics - we can work purely in terms of coordinates, vectors, or Lagrangians, but the epistemological foundations are unchanged, and rest on the symmetries identified for inertial coordinates. When we use other coordinates, we just make the necessary adjustments, and then if we wish we can use the tensor formalism to sweep them all together for typographical convenience, but the foundations are unchanged.

You posed the question backwards. The Doppler shift is the direct observable; the relative velocity v is a physical interpretation of what the observable means.

That's simply not true. The concept of relative velocity between two objects has a perfectly valid operational meaning, independent of the Doppler shift, so it is perfectly valid to ask what the Doppler shift is for a given relative velocity. Indeed this is the kind of thing one checks to test for Lorentz invariance. That's why I suggested you think about how you would test for that.

We define relative speed in terms of observed Doppler shift.

No! That's the heart of your problem. We do NOT define relative speed in terms of the observed Doppler shift. Relative speed has a perfectly good operational meaning independent of the Doppler shift. The empirical fact that the Doppler shift for a given velocity matches the relativistic prediction is an empirical test of Lorentz invariance. This is NOT tautological.

The same way it's already been tested. I don't think we disagree at all about experimental results; we're talking about the logical structure of the theory we use to describe them.

The reason I suggested you think about how YOU would test for Lorentz invariance is because it would force you to recognize that, according to your current understanding, it cannot be tested! For example, you believe that relative velocity has no physical meaning other than the Doppler shift it produces, and so the relationship between speed and Doppler shift is simply a matter of definition, hence cannot be tested. But that is completely wrong. Lorentz invariance is not tautological. The foundations of special relativity can't be understood without understanding this.

 

[PAllen]

People were taking measurements and doing experiments long before the concept of coordinate systems was invented. Nature has no coordinates. A so called operational coordinate system can be have the coordinates removed without essential loss.

Measuring speed without Doppler (e.g. taking a reference object (ruler) and laying it multiple times to measure a distance and then using a watch to measure flight times for baseballs thrown from start to finish) does not establish coordinates. The facts of the situation remain whether or not you invent coordinates for the description.

 

[PAllen]

As for testing for Lorentz invariance, you don't. You t test for consequences of a mathematical model that incorporates Lorentz invariance, or (equivalently) flat pseudo-riemannian geometry. That is, you perform the MM experiment, you test the decay rate of muons in a ring, you test for muons arriving on the ground combined with other tests for cosmic rays. You test how you need to adjust GPS systems.

 

[PAllen]

I'll offer an analogy here. Is it necessary to understand flat plane geometry, and verify this geometry, to introduce cartesian coordinates? Euclid's coordinate free understanding was flawed and inadequate?

Similarly, what we verify (to the extend GR allows it locally) flat pseudo-riemannian geometry plus correspondence rules between geometric features of this manifold and observations. Frames and coordinates are just one tool that can be used.

 

[WannabeNewton]

I refer again to section 13.1 in Wald. It should make things clearer with regards to special covariance and what Sam mentioned in post #39 and related posts.

 

[PeterDonis]

orthonormality is based on the isotropy of inertia (just as with an inertial coordinate system).

That doesn't mean that isotropy of inertia requires defining an inertial coordinate system.

This is (for example) ultimately how we decide what time direction is orthonormal to the space directions for any given state of motion.

Sure, but again, this can be done without defining inertial coordinates. The physical property of isotropy of inertia is distinct from the abstract construction of inertial coordinates.

You want to invoke things like orthonormal 4-tuples without acknowledging that it is tantamount to an inertial coordinate system

Yes, because it isn't tantamount to an inertial coordinate system. See above.

orthonormality has non-trivial physical significance, based on empirical phenomena that could (a priori) have been otherwise.

I agree with that, but I don't agree that it requires defining an inertial coordinate system.

It's only a geometric object if we have a metric

No, that's not true. You can define geometric objects without a metric; but you can't inter-convert vectors and covectors (or 1-forms) without a metric, which means that without a metric, you can only contract a vector with a 1-form; you can't contract vectors with vectors or 1-forms with 1-forms. Similarly for higher-order tensors and forms.

the metric must be operationally defined in order to have physical meaning.

Agreed; this is true of any geometric object. But you don't need to define coordinates to operationally define geometric objects, so the circularity you speak of is not there.

We can't do anything (much) with just the generic idea of a metric. To make actual predictions, we need a definite metric, and although it can be expressed in terms of different coordinate systems

Or it can be expressed without any coordinates at all.

We need some way of translating observable facts so as to constrain and determine this metric sufficiently to answer questions.

Agreed.

Ultimately we can only do this by observing the behavior of phenomena, and establishing the coordinated measures that collectively amount to a coordinate system (although of course it need not be expressed formally as a coordinate system).

But, once again, you can define and express observable, measurable numbers without defining coordinates. Coordinates make all this easier, but that's not the same as saying they're required.

Ah yes, MTW, the apostles of the coordinate-free faith.

That example (paragraph 2.8) is a classic for how it tacitly smuggles in all the information, not least when they blithly equate the wavelength ratio with the energy ratio, and when they claim the magnitude of the "ordinary velocity" of the rim is unchanging - which of course is only true if the centrifuge is at rest in an inertial coordinate system

Huh? Once again, ordinary velocity can be defined entirely in terms of observables; coordinates are not needed.

and only if we understand what "ordinary velocity" means, which they aren't even entitled to talk about in the absence of coordinates

Same comment. I already addressed this point in a previous post. If we're just going to have to agree to disagree, that's fine, but you can't respond to specific examples by just repeating your assertions without supporting argument.

People who think they are dispensing with inertial coordinates never really are.

I think you are confusing inertial coordinates with the physical properties that make inertial coordinates useful.

The concept of relative velocity between two objects has a perfectly valid operational meaning, independent of the Doppler shift

Then what is it? Be specific.

No! That's the heart of your problem. We do NOT define relative speed in terms of the observed Doppler shift.

Maybe you don't, but that doesn't mean it can't be done. Once again, can you give a specific alternative?

Also, supposing that you can give a specific alternative, why should your alternative be privileged over mine? If there are two alternative ways of operationally defining what "relative velocity" means, then we have an empirical question: do they always give the same answer? If they do, that's an interesting physical fact that can be investigated further.

you believe that relative velocity has no physical meaning other than the Doppler shift it produces

Where did I say that? I said relative speed *can* be defined in terms of Doppler shift; I did not say it *has* to be. Perhaps my remarks just above will help to clarify where I'm coming from.

and so the relationship between speed and Doppler shift is simply a matter of definition, hence cannot be tested.

But Lorentz invariance is not just a matter of defining relative velocity, whether in terms of Doppler shift or anything else.

To put it another way, the argument you give here, if it were valid, would prove too much. Suppose you come up with an alternative operational definition of relative velocity, as I asked for above; why wouldn't the argument you give here apply equally well to that definition?

 

[WannabeNewton]

You want to invoke things like orthonormal 4-tuples without acknowledging that it is tantamount to an inertial coordinate system...

I must disagree with this to some extent. I guess it depends on how strongly you use the word tantamount. The local Lorentz frames are in and of themselves just special orthonormal frames defined at a given event that physically represent a measuring apparatus consisting of a clock and three mutually perpendicular meter sticks; that they can be used to define a locally inertial coordinate system about a neighborhood of the event is a non-trivial consequence of the exponential map in semi-Riemannian geometry. So I guess you could say one follows from the other but I wouldn't say they were "essentially equivalent".

 

[Samshorn]

People were taking measurements and doing experiments long before the concept of coordinate systems was invented. Nature has no coordinates.

They were using the inertial measures of space and time, even though they didn't realize it until Galileo came along. Equal action and reaction has always been intuitive, as has the relativity of inertial systems of reference. (We can juggle just as well on a ship as on land.) The point is not whether someone meticulously assigns coordinates to every event, it is that they use the meanings of length, time, and simultaneity implicit in inertial coordinates.

...taking a reference object (ruler) and laying it multiple times to measure a distance and then using a watch to measure flight times for baseballs thrown from start to finish) does not establish coordinates.

Well, assigning distances to events using rods in co-moving inertial motion, and assigning the times of events with the synchronization such that the laws of mechanics hold good (equal action and reaction), surely qualifies as using inertial coordinates. Whether or not you explicitly assign coordinates to every event in the universe, or just to a few events of interest, is irrelevant. You are still using the measures of space, time, and motion represented by inertial coordinates.

As for testing for Lorentz invariance, you don't...

I disagree. The experiments you mentioned, along with many others up to the present day, are tests of Lorentz invariance. Physics is not tautological.

I'll offer an analogy here. Is it necessary to understand flat plane geometry, and verify this geometry, to introduce cartesian coordinates?

To the extent that Euclidean geometry is a physical theory (as opposed to just an abstract axiomatic structure), the relevant analogy is the isotropy of spatial orientations of stable physical objects. The principle of relativity here is that the laws governing the shapes and sizes of physical objects are invariant under changes in orientation and translation. These are the symmetries (see Klein's Erlangen program) that are exhibited by physical objects in Euclidean relativity, but of course we can't say a "solid" object has the same equilibrium length when oriented in any direction, all we can say is that it covers the same number of rulers when orientated in any direction. Hence all we really know is that, however the object's length is affected by orientation, the ruler is affected in exactly the same way... and so is everything else. Every measurement is really just a comparison of something with something else. Again, it isn't necessary to completely populate the entire space with articulated coordinates to be using different orientations to express the physical symmetry. In spacetime, those different orientations correspond to inertial frames.

The problem with your outlook is that you just have one hand clapping. This is what leads you to say thing like "As for testing for Lorentz invariance, you don't...". The reason you say that (unaware of all the tests of Lorentz invariance that are carried out), is because according to your view, we can't test Lorentz invariance, because you're just clapping with one hand... you don't understand the significance of the inertial measures of space and time intervals.

The physical property of isotropy of inertia is distinct from the abstract construction of inertial coordinates.

Well, it's the property on which inertial coordinates are constructed, establishing a unique simultaneity, that can then be compared with the simultaneity given by isotropic light speed. These are the two hands clapping.

Once again, ordinary velocity can be defined entirely in terms of observables; coordinates are not needed.

The concept of speed has well-established meaning in terms of inertial measures of spatial distances and time intervals. These are the measures corresponding to inertial coordinate systems. It doesn't matter if you explicitly populate the entire space and time with coordinates. The point is that you are using the measures of space and time corresponding to an inertial coordinate system (just as MTW were in their centrifuge, which of course was utterly trivial and didn't require any quantitative reasoning at all to recognize that there would be no Doppler shift).

If there are two alternative ways of operationally defining what "relative velocity" means, then we have an empirical question: do they always give the same answer? If they do, that's an interesting physical fact...

Yes, that interesting physical fact is called Lorentz invariance. Remember, there was already a pre-existing definition of "relative velocity" (i.e., distance divided by time, both defined in terms of coordinates in which the equations of mechanics hold good), and we can then ask if an object moving with a velocity v exhibits the Doppler shift predicted by the Lorentz transformation.

I said relative speed *can* be defined in terms of Doppler shift; I did not say it *has* to be.

The point is that if you define the (one-dimensional) speed of an object as whatever it must be to satisfy the relativistic Doppler equation, then you are simply defining the relativistic Doppler equation to be valid... it is no longer a falsifiable proposition... but we know it IS a falsifiable proposition. The reason we know this is because the concept of speed has meaning independent of Doppler shift. When we test Lorentz invariance, we use that "ordinary speed" and check to see if the Doppler equation gives the observed shift. This shows the crucial significance of the definition of ordinary speed, which is nothing but the speed given by inertial measures of space and time.

Suppose you come up with an alternative operational definition of relative velocity, as I asked for above; why wouldn't the argument you give here apply equally well to that definition?

Again, the physical meaning of (for example) the relativistic Doppler equation is that it relates the frequency shift to the ordinary speed, and this speed is not defined circularly in terms of the Doppler shift, it is defined based on the inertial measures of space and time, which we conveniently refer to as "inertial coordinate systems". (By the way, there are other problems with using Doppler to define speeds in more than one dimension, which leads to the absoluteness of rotation, another crucial aspect of inertial measures.)

 

[Dale]

The same way it's already been tested. I don't think we disagree at all about experimental results; we're talking about the logical structure of the theory we use to describe them.

I am not aware of any tensor-based or coordinate-independent test theory of SR. Are you?

I think that one could, in principle, be developed. But to my knowledge it hasn't yet.

 

[PAllen]

It seems part of this disagreement is terminological. To me, coordinates are system of labels covering a region; in contrast to measurements of distances, time intervals, and angles which I view as invariants. To me, stating that several instruments are each feeling no accelerations (measured with ideal accelerometers) and collectively maintaining fixed relative positions is part of the recipe for measurement; placing this in an inertial frame or coordinates is simply one way of describing the set up. Stating that a measurement comes out the same wherever, whenever, at any relative speed to something else, and in what orientation you do it is describing an invariant symmetry. [In this sense, I would correct my statement that you can't verify Lorentz invariance, meant as this collection of symmetries.] So, basically, all the things I define as geometric or physical invariants independent of coordinates or frames, Samshorn is bundling into an (to me) extended concept inertial frames.I don't see a constructive way to continue with such different definitions. I do see that the value in my approach is to separate concepts I consider invariant and coordinate independent from features of particular systems of labels, because I have seen much confusion in this area.

 

[PeterDonis]

Well, it's the property on which inertial coordinates are constructed, establishing a unique simultaneity, that can then be compared with the simultaneity given by isotropic light speed.

Yes, you can, *if you want to*, construct inertial coordinates using this property. That doesn't mean you *have* to do so, or that the coordinates are identical with the property. As I said before and as PAllen has pointed out, you appear to be conflating "inertial coordinates" with "physical properties that can be used to construct inertial coordinates".

The concept of speed has well-established meaning in terms of inertial measures of spatial distances and time intervals.

This is a fact about history, not physics. The fact that relative velocity was first defined in terms of these measures, and only later related to the Doppler shift, does not mean the Doppler shift is less fundamental.

The point is that if you define the (one-dimensional) speed of an object as whatever it must be to satisfy the relativistic Doppler equation, then you are simply defining the relativistic Doppler equation to be valid... it is no longer a falsifiable proposition... but we know it IS a falsifiable proposition. The reason we know this is because the concept of speed has meaning independent of Doppler shift.

But, as I said before, I can equally well turn this argument around: "the point is that if you define the speed of an object as whatever it must be to satisfy the inertial measures of distance and time, then you are simply defining those inertial measures to be valid; it is no longer a falsifiable proposition that relative velocity, so defined, is the "v" that appears in the Lorentz transformation equations. But we know it IS a falsifiable proposition. The reason we know this is because the concept of speed has meaning independent of those particular measures."

Once again, you appear to be privileging your definition of "speed" simply because it happened to be the one that was discovered first. But physically speaking, the fact that that "speed" happens to be equal to the "v" that appears in the Lorentz transformations is just as much a contingent, falsifiable proposition as the fact that the observed Doppler shift happens to be just right to make the relativistic Doppler equation valid. It could have turned out that the "speed" defined by inertial measures satisfied some other transformation equation, such as the Galilean transformation. The fact that it didn't is an empirical fact, not an a priori definition of speed.

 

[PeterDonis]

I am not aware of any tensor-based or coordinate-independent test theory of SR. Are you?

Sure, just write down all the equations the same way you can in GR, using only coordinate-free tensor expressions, and use the metric of Minkowski spacetime.

Or do you mean by "test theory" something like the Cartan geometric formulation of Newtonian gravity? In other words, casting Galilean spacetime in terms of coordinate-free tensor equations? I don't know that that has been done specifically, but I don't see why it couldn't be.

 

[Samshorn]

It seems part of this disagreement is terminological. To me, coordinates are system of labels covering a region; in contrast to measurements of distances, time intervals, and angles which I view as invariants.

Spatial distances and temporal intervals (and speeds, computed as the ratios of those two things), as well as angles are all frame dependent things (see length contraction, time dilation, and angular aberration), and when we refer to or make a measurement of these things, we are typically (implicitly or explicitly) invoking the measures corresponding to a particular system of inertial coordinates (just as MTW did in their allegedly "coordinate free" "calculation" - which wasn't coordinate free and wasn't a calculation).

Now, there is a sense in which these things ARE "invariant", once they are fully specified. There's a difference between (1) the spatial extent of an object, and (2) the spatial extent of an object in terms of a particular well-defined system of inertial coordinates. Item (1) is ambiguous and frame dependent, but item (2) is "invariant", because it includes the stipulation of both the object and the operational meaning of spatial extent that we intend (assuming you understand how inertial coordinate systems are operationally defined) - and that meaning is not defined circularly. This just highlights the fact that frame-dependent quantities are not physically meaningless. Ultimately every measurement is simply a comparison of something with something else.

For example, "the one-way speed of light" is ambiguous, but "the one-way speed of light in terms of a system of coordinates in which the equations of mechanics hold good" is a matter of empirical fact that can be tested and measured. It is not circular or tautological. It's vital to recognize and understand this, because it represents the empirical content of special relativity. Too often special relativity is presented as if it was just a bunch of unfalsifiable conventions, precisely because people don't grasp the independent physical significance of the inertial measures of space and time.

To me, stating that several instruments are each feeling no accelerations (measured with ideal accelerometers) and collectively maintaining fixed relative positions is part of the recipe for measurement; placing this in an inertial frame or coordinates is simply one way of describing the set up.

You're leaving out simultaneity. The crucial point is that the "coordinates in which the equations of mechanics hold good" (including equal action and reaction) entail a unique simultaneity, which is the basis for all measures of space, time, motion, angles, etc., within a given frame. Any time you refer to or measure those things, you are using the inertial coordinates implicit for that frame.

 

[WannabeNewton]

The whole point of special covariance is that if a family of observers make a complete set of measurements of the components of tensor fields on space-time (states of space-time) using the their measuring apparatuses (orthonormal frame field) and we act on the tensor fields by a diffeomorphism then the components of the images will correspond to a complete set of measurements made by a new family of observers (gotten by acting the diffeomorphism on the original orthonormal frame field) if and only if the diffeomorphism is an isometry. In the case of SR, these will be the representations of the proper Poincare group and tells us the above about the locally physically measurable quantities (components of tensor fields on Minkowski space-time).

 

[Samshorn]

But, as I said before, I can equally well turn this argument around: "the point is that if you define the speed of an object as whatever it must be to satisfy the inertial measures of distance and time, then you are simply defining those inertial measures to be valid; it is no longer a falsifiable proposition that relative velocity, so defined, is the "v" that appears in the Lorentz transformation equations. But we know it IS a falsifiable proposition. The reason we know this is because the concept of speed has meaning independent of those particular measures."

You got mixed up in your turn-around. The correct turn-around statement is this: If we define inertial coordinate systems as "systems of space and time coordinates in terms of which the mechanical inertia of every object is homogeneous and isotropic", then the homogeneity and isotropy of mechanical inertia is true by definition, i.e., not falsifiable. Well, this is perfectly correct (and well known - it was pointed out centuries ago that Newton's "laws" really constitute the definition of inertial coordinate systems), and indeed we could construct infinitely many more such tautologies. For example, if I define 'samshorn coordinates' as systems in terms of which all objects move in circles, then it is formally tautological that all objects move in circles in terms of samshorn coordinates. The difference is that, as an empirical matter, no samshorn coordinates exist, so the proposition has no applicability.

However, miraculously, there actually DO exist inertial coordinate systems (as defined). These kinds of definitions are hugely over-specified, and we have no right to expect that any single system of coordinates (let alone a whole family of them) could exist in which the mechanical inertia of every object is homogeneous and isotropic, any more than we should expect there to exist coordinates in terms of which every object moves in a circle. But based on enormous amounts of experience and observation, it appears (empirically) that inertial coordinates do exist - i.e., physical phenomena do exibit that amazing degree of coherence and uniformity. That coherence and uniformity, represented by the existence of inertial coordinate systems, is what makes the science of mechanics possible, and of course, living here on the moving Earth, we have always exploited this wonderful property to define our measures of space, time, motion, and simultaneity in terms of inertial coordinates (even before we realized we were doing it).

Now, when we examine the Doppler effect (for example), we find an equation that relates speed (in the inertial sense) to the shift in frequency, and we can test this equation to see how these frequency shifts of characteristic phenomena of emitting objects fits into our miraculous coherent and uniform physics of mechanics. Your position seems to be that the meaning of this equation has nothing to do with the ordinary inertial meaning of speed. You contend that we should simply interpret the Doppler equation as the definition of speed. My contention is that you are thereby discarding the entire physical meaning and significance of that equation. The content of that equation - and all the others representing Lorentz invariance - is precisely to show how the phenomena of emitting entities relates to the inertial measures of space and time.

 

[PeterDonis]

The correct turn-around statement is this: If we define inertial coordinate systems as "systems of space and time coordinates in terms of which the mechanical inertia of every object is homogeneous and isotropic", then the homogeneity and isotropy of mechanical inertia is true by definition, i.e., not falsifiable.

You are basically saying that "mechanical inertia" is coordinate-dependent. I'm not sure that's correct, but let's assume it is for the sake of argument. Then my response is that mechanical inertia, as you've defined it, is not a physical property! It can't be, because physical properties must be expressible in terms of invariants, and mechanical inertia, by your definition, is not; it depends on the coordinates you choose.

To put it another way: you're basically saying that inertial mass is not a scalar; its value will be different from its "inertial value" if I measure it relative to non-inertial coordinates. Again, I'm not sure that's correct, but let's assume it is for the sake of argument. Then inertial mass, as you've defined it, is not the real physical observable; the real physical observable would be a scalar describing the contraction of some geometric object describing "inertial mass" with one of the vectors in the 4-tuple describing the reference frame. Changing to non-inertial coordinates would change the 4-tuple, and therefore would change the contraction (assuming the "inertial mass" geometric object remained fixed).

 

[Samshorn]

You are basically saying that "mechanical inertia" is coordinate-dependent. I'm not sure that's correct...

One of the first and most important things one learns about special relativity is that energy (all forms of energy) has inertia - and this includes kinetic energy, which of course is frame dependent. Hence inertia is unavoidably frame dependent. This is the very cornerstone of special relativity.

...but let's assume it is for the sake of argument. Then my response is that mechanical inertia, as you've defined it, is not a physical property! It can't be, because physical properties must be expressible in terms of invariants, and mechanical inertia, by your definition, is not; it depends on the coordinates you choose.

So, you assert that kinetic energy is not a "physical property", because it is frame-dependent. I would say it differently: Kinetic energy is a frame-dependent physical property.

You're basically saying that inertial mass is not a scalar...

No, I'm saying all forms of energy - including kinetic energy - have inertia. (I wish I could take credit for this insight, but it's actually the well-known cornerstone of special relativity.) One consequence of this is that if we construct two inertial coordinate systems using mechanical inertia (i.e., such that mechanical inertia is homogeneous and isotropic), we get two different simultaneities. Of course, they are the very same simultaneities we get for light speed to be homogeneous and isotropic in the two frames. This works only because energy has inertia.

It might be helpful for you to think about how you would actually define an inertial coordinate system mechanically. This would make it clear how the inertia of (kinetic) energy comes into play. Hopefully when you've satisfied yourself that energy does indeed have inertia, you can make it past the first paragraph of my previous post. I think the next two paragraphs of that post explained things fairly well.

 

[PeterDonis]

One of the first and most important things one learns about special relativity is that energy (all forms of energy) has inertia - and this includes kinetic energy, which of course is frame dependent. Hence inertia is unavoidably frame dependent.

This is a good point, but note that it is *not* a cornerstone of Newtonian physics. In other words, "energy has inertia" is an empirical fact that helps to distinguish Newtonian/Galilean relativity from Lorentzian/Einsteinian relativity.

So, you assert that kinetic energy is not a "physical property", because it is frame-dependent. I would say it differently: Kinetic energy is a frame-dependent physical property.

Yes, I should have been more precise. I think I said in a previous post that total energy is the contraction of the object's 4-momentum with the observer's 4-velocity; kinetic energy is total energy minus the invariant length of the object's 4-momentum. Total energy and kinetic energy are therefore observer-dependent, yes; but we can write down an invariant expression for "energy of object X as measured by observer O", so in that sense "energy" is a physical property, yes.

However, I don't think any of this requires that the mechanical definition of relative speed be privileged over the Doppler definition.

 

[Samshorn]

I don't think any of this requires that the mechanical definition of relative speed be privileged over the Doppler definition.

You keep talking about one definition being privileged over another, but it isn't a question of privilege, it's a question of what the Doppler formula means. It means that if you plug in the ordinary speed v (based on the inertial definition), you get the relativistic Doppler shift. That's the non-trivial physical fact that the formula is expressing. If you refuse to recognize that the parameter v in that formula represents the inertial speed, then the formula doesn't mean anything, i.e., it doesn't connect something to something else. It's one hand clapping.

Your position seems to be that the meaning of v in the Doppler equation need not have anything to do with the ordinary inertial meaning of speed. You contend that we can just as well interpret the Doppler equation itself as the definition of the parameter v appearing in it. My contention is that you are thereby discarding the entire physical meaning and significance of that equation. The function of that equation is precisely to show how the frequencies of emitting entities relate to the inertial measures of space and time.

I hesitate to mention it, because it may just divert attention from the main point, but it is actually somewhat relevant to the main point: In more than one dimension there isn't even a one-to-one correspondence between speed and Doppler shift. For example, if you follow an equi-angular spiral path away from a source of light, with the right combination of angle and speed, you will have no Doppler shift at all, even though you obviously have speed (both radial and tangential) relative to the source. Of course, when we say this, we are talking about speeds defined in the inertial sense. We can't encode the full range of possible (3-dimensional) motions in terms of the Doppler shift from some emitter. (If you posit three or four stategically placed emitters, you are just constructing a coordinate system.)

Note that we called this motion "spiral", because we are changing the angular orientation of the line between the source and the object... but why can't we just define that line to be stationary, so that the object is simply moving away radially? Well, because the inertial sense of motion entails an absolute sense of rotation. So, again, the very statement of the problem unavoidably involves the use of inertial definition of motion. The same applies to the MTW exercise you cited, in which they smuggled in the "ordinary velocity v" without even blushing. This all just shows why we define motions in the sense of inertial coordinates, and we then determine formulas giving the Doppler shift for any specified motions. It wouldn't make sense to try to do the reverse, e.g., to try to infer the spiral motion from the (absence of) Doppler shift.

 

[PeterDonis]

it's a question of what the Doppler formula means. It means that if you plug in the ordinary speed v (based on the inertial definition), you get the relativistic Doppler shift. That's the non-trivial physical fact that the formula is expressing.

Yes, but that physical fact by itself doesn't tell you whether you should interpret it as telling you that, oh, look, the observed Doppler shift just happens to be exactly equal to what the formula tells you when you plug in speed v; or telling you that, oh, look, the speed v just happens to be exactly what you would expect when you plug the Doppler shift into the (inverted) formula.

Your position seems to be that the meaning of v in the Doppler equation need not have anything to do with the ordinary inertial meaning of speed.

No, my position is that the physical fact the formula is expressing can be interpreted in either direction, so to speak. It's telling you about a connection between two different sets of phenomena. It's not telling you which set of phenomena is "more fundamental". That's a matter of interpretation.

 

[Samshorn]

Yes, but that physical fact by itself doesn't tell you whether you should interpret it as telling you that, oh, look, the observed Doppler shift just happens to be exactly equal to what the formula tells you when you plug in speed v; or telling you that, oh, look, the speed v just happens to be exactly what you would expect when you plug the Doppler shift into the (inverted) formula.

In both of the "alternatives" you mentioned, the equation is empirically valid only if v equals the inertial speed, which is the point I've been making: The formula refers to inertial speed or else it's meaningless.

No, my position is that the physical fact the formula is expressing can be interpreted in either direction, so to speak. It's telling you about a connection between two different sets of phenomena. It's not telling you which set of phenomena is "more fundamental". That's a matter of interpretation.

I don't think it's a question of whether one thing is "more fundamental" than another (whatever that might mean). The point is just that both are physically meaningful, and the task of the theory is to describe how they are related. That's why we shouldn't say special relativity has nothing to do with inertial coordinate systems or inertial measures of space and time (or with the Lorentz transformations that relate them to each other). It has everything to do with these things.

As to why we ordinarily conceive of phenomena as existing in space and time, rather than in some abstract "Doppler realm", I think that's a complicated subject. There are always naive positivists urging us to shed our hide-bound notions of a common space and time, and just focus on the raw uninterpreted sense impressions impinging on our own individual world line. The natural tendency of this line of thought is toward solipsism, but even if you succeed in avoiding that trap, most people find that there are good reasons for retaining the conceptual framework of space and time. (I mentioned some in a previous message.) On the other hand, quantum phenomena for n particles seem to reside in 3n-dimensional phase space, suggesting that our concept of 3+1 dimensional spacetime is not fundamental. Nevertheless, we somehow still make the conversion back to space-time representations for most purposes.

 

[bobc2]

...There are always naive positivists urging us to shed our hide-bound notions of a common space and time, and just focus on the raw uninterpreted sense impressions impinging on our own individual world line. The natural tendency of this line of thought is toward solipsism, but even if you succeed in avoiding that trap, most people find that there are good reasons for retaining the conceptual framework of space and time. (I mentioned some in a previous message.)...

Well said.

 

[PeterDonis]

In both of the "alternatives" you mentioned, the equation is empirically valid only if v equals the inertial speed

Yes, but "inertial speed" means "the actual measured speed according to a given procedure". You can describe that procedure and its results without defining inertial coordinates.

That's why we shouldn't say special relativity has nothing to do with inertial coordinate systems or inertial measures of space and time (or with the Lorentz transformations that relate them to each other). It has everything to do with these things.

Once again, you are conflating inertial coordinates with the physical properties and measurements that make inertial coordinates useful. SR does have everything to do with those physical properties and measurements, yes. But you can describe them without defining inertial coordinates.

As to why we ordinarily conceive of phenomena as existing in space and time, rather than in some abstract "Doppler realm", I think that's a complicated subject.

Yes, it is, because it's not just about physics; it's about our cognitive systems, which are much more complicated than the simple physical systems we're talking about here.

There are always naive positivists urging us to shed our hide-bound notions of a common space and time, and just focus on the raw uninterpreted sense impressions impinging on our own individual world line.

I have not said anything like this. Once again, you're conflating coordinates with physical properties that are conveniently described using coordinates. Spacetime, as a geometric object, is certainly the natural outcome of reconciling our common notions of space and time with the other physical facts we have discussed. But spacetime can be described without coordinates. That's all I'm saying. That's not the same as saying spacetime doesn't exist, only our sense impressions do.

 

[Samshorn]

Yes, but "inertial speed" means "the actual measured speed according to a given procedure". You can describe that procedure and its results without defining inertial coordinates.

If someone defines a speed (for example) using a procedure operationally equivalent to defining an inertial coordinate system and dividing the space difference by the time difference, then I'd count that as using an inertial coordinate system, which is to say, using inertial measures of space, time, speed, angles, etc. For example, the MTW exercise you cited as a coordinate-free "calculation" used an inertial coordinate system by referring to the "ordinary speed v".

Once again, you are conflating inertial coordinates with the physical properties and measurements that make inertial coordinates useful.

No, I'm conflating the use of inertial coordinates with the use of measures of distance, time, speed, angles, etc., based on the concept of inertial coordinates. If, for example, someone says an object is moving at speed v, and they mean this as the ordinary speed in terms of an inertial coordinate system, then I would say they have invoked an inertial coordinate system. (The v in the Doppler formula is just such a speed, and hence it refers to inertial coordinates.)

SR does have everything to do with those physical properties and measurements, yes. But you can describe them without defining inertial coordinates.

If someone asks you what special relativity predicts for the Doppler shift of a light source moving away from us at the speed v, where v is an "ordinary speed" defined in terms of inertial coordinates, how would your application of coordinate-free reasoning erase the fact that what you're doing is explicitly answering a question whose parameters are defined in terms of inertial coordinates?

Spacetime can be described without coordinates. That's all I'm saying.

Ohanian and Ruffini had it right: "We must not forget that the physicist who wishes to measure a tensor has no choice but to set up a coordinate system, and then measure the numerical values of the components. Thus, to carry out the comparison of theory and experiment, the physicist cannot ultimately avoid the language of components; only a pure mathematician can adhere exclusively to the abstract coordinate-free language of differential forms..." You see, this is the point: The epistemological foundations of a physical theory rest entirely on "the comparison of theory and experiment", and this decisive link is provided by coordinates. Without this link, any formal mathematical expressions are devoid of physical content.

 

[Dale]

Sure, just write down all the equations the same way you can in GR, using only coordinate-free tensor expressions, and use the metric of Minkowski spacetime.

No, if you use the metric of Minkowski spacetime then you are already assuming SR and not making a test theory for SR.

Or do you mean by "test theory" something like the Cartan geometric formulation of Newtonian gravity? In other words, casting Galilean spacetime in terms of coordinate-free tensor equations?

In order to test a theory you cannot assume it, so the best approach is to assume a test theory. A test theory is a general theory which has a set of one or more unknown parameters. Various competing theories (such as SR or Newtonian physics) are then reproduced through specific choices of the unknown parameters. You then propose an experiment to put constraints on the unknown parameters and see how closely they match the parameters corresponding to the various theories.

So, in this case, a coordinate-free test theory would be one which reproduces either Minkowski spacetime or Galilean spacetime with some set of tensors and scalars.

I don't know that that has been done specifically, but I don't see why it couldn't be.

I also don't see why it couldn't be, but as far as I know it has not been done, so I would be reluctant to make claims about it.

 

[Dale]

Ohanian and Ruffini had it right: "We must not forget that the physicist who wishes to measure a tensor has no choice but to set up a coordinate system, and then measure the numerical values of the components. Thus, to carry out the comparison of theory and experiment, the physicist cannot ultimately avoid the language of components; only a pure mathematician can adhere exclusively to the abstract coordinate-free language of differential forms..."

Ohanian and Ruffini are wrong on this point ("no choice but to set up a coordinate system"). You can set up a set of basis vectors without setting up a coordinate system. Components are then contractions with one of the basis vectors, which is still coordinate-free. So just because you are dealing with components doesn't mean that you are dealing with coordinates. Their conclusion ("no choice ...") doesn't follow from their argument ("cannot ultimately avoid ... components").

Coordinates always imply a unique vector field called the coordinate basis, but a basis does not imply a unique coordinate system.

 

[Phy_Man]

Ohanian and Ruffini are wrong on this point ("no choice but to set up a coordinate system"). You can set up a set of basis vectors without setting up a coordinate system. Components are then contractions with one of the basis vectors, which is still coordinate-free. So just because you are dealing with components doesn't mean that you are dealing with coordinates. Their conclusion ("no choice ...") doesn't follow from their argument ("cannot ultimately avoid ... components").

Coordinates always imply a unique vector field called the coordinate basis, but a basis does not imply a unique coordinate system.

Ummmm, nope! I'm afraid that you missed a very fundamental fact here. Ohanian and Ruffini are absolutely correct. They're experts in their field and know precisely what they're talking about. Setting up a system of basis vectors is identically the same thing as setting up a coordinate system. Contracting components on a basis is just another name for measuring components. Dealing with components is identical to dealing with a coordinate system. From your response it appear as if you might have a flawed notion of what a coordinate system is.

 

[Dale]

Setting up a system of basis vectors is identically the same thing as setting up a coordinate system.

See the attached image for a system of basis vectors. What are the coordinates of the blue point?

 

[Phy_Man]

See the attached image for a system of basis vectors. What are the coordinates of the blue point?

I'm sorry my good man but I don't see a blue point. I assume that you understand that when I said that setting up a coordinate system is identical to settiing up a basis that it can't be taken to mean that a set of basis vectors is a coordinate system, right?

Regarding your question, I assume you meant to place a point in that diagram somewhere to represent the position vector which is a displacement vector from a point chosen as the origin? While you're at it please make it clear as to whether you're asking me for the components of the displacement vector (which is what a position vector is) from the origin that the point represents or whether you have something else in mind. Thank you.

When someone sets up a basis then wants to express the position vector in terms of that basis they have to locate the two points which define the displacement vector; one point represents the point represented by the origin and the point of interest. When you tell me that there is a point in there and I should tell you its coordinates then what you're telling me is that you have chosen a point to serve as the origin. If not then you haven't mentioned a displacement vector and I have no response as a result.

If you missed that point then don't feel bad. It's no big deal. A lot of people seem to forget what it means to represent a point or speak of its coordinates. I.e. if you wish to speak of coordinates then what you've implied by your question is that you have chosen a point to serve as the origin and then the point is the vector displacement from the reference point (i.e. origin) to the point in question.

Or perhaps you thought you had me? Nope. Sorry. :D