Acceleration and velocity: Newtonian versus relativistic interpretation.

[PeterDonis]

How does the concept of a rotating / non-rotating frame carry over to general relativity?

You might want to read up on http://en.wikipedia.org/wiki/Fermi%E2%80%93Walker_transport" .

 

[matheinste]

As a matter of principle, velocity with respect to SR-spacetime does not enter SR. (And for good measure: no velocity vector can be assigned to one part of SR-spacetime moving with respect to another part of spacetime.)

It seems to me that the above principle is necessary and sufficient so segregate aether notions from special relativity.

Cleonis

Bearing in mind that aether dragging was disproved experimentally and discounted many years ago, and movement relative to the aether is undetectedable, if we replace the words "SR-spacetime" by "the aether", what difference does it make.

Matheinste.

 

[Cleonis]

The concept of 4-velocity as the tangent vector to a worldline at an event does apply in SR (as does 4-acceleration as the derivative of that tangent vector with respect to proper time- [...]) I could easily see an aether theorist arguing that 4-velocity is "velocity with respect to spacetime". So I'm not sure your suggestion will work.

Well, I expect it will work, in the sense that I think the reasoning is sound. (Still, I haven't dug as deep as I could, so I won't venture further than 'I expect'.)

In all there is the two-step cascade:
- Velocity is time derivative of position
- Acceleration is time derivative of velocity.

We have that 4-position is not position with respect to spacetime; as a matter of principle an absolute reference of position does not enter special relativity.

Cleonis

 

[Cleonis]

Bearing in mind that aether dragging was disproved experimentally and discounted many years ago, and movement relative to the aether is undetectedable, if we replace the words "SR-spacetime" by "the aether", what difference does it make.

There is a difference.

General remark about Lorents ether theory: there is consensus on the following issue: Lorentz ether theory matches all of the SR predictions. In that sense Lorentz ether theory and special relativity are experimentally indistinguishable. Yet the scientific community regards special relativity as a theory that has superseded Lorentz ether theory.

The reason for that is that in science experimental outcome is not the only factor that influences commitment to a particular theory. The scientific community is committed to special relativity because it is a far more economical theory.

Lorentz theory is regarded as requiring ad hoc hypotheses to keep it going, and the expectation is that if Lorentz ether theory is probed deeper more and more situations will be encountered that require an ad hoc hypothesis to patch it up. Conversely, in the case of special relativity the expectation is that its small set of postulates is exhaustive; so far no ad hoc hypothesis has been needed, and that status is expected to remain so.

This economy is why special relativity has become the dominant theory.

Cleonis

 

[matheinste]

There is a difference.

General remark about Lorents ether theory: there is consensus on the following issue: Lorentz ether theory matches all of the SR predictions. In that sense Lorentz ether theory and special relativity are experimentally indistinguishable. Yet the scientific community regards special relativity as a theory that has superseded Lorentz ether theory.

The reason for that is that in science experimental outcome is not the only factor that influences commitment to a particular theory. The scientific community is committed to special relativity because it is a far more economical theory.

Lorentz theory is regarded as requiring ad hoc hypotheses to keep it going, and the expectation is that if Lorentz ether theory is probed deeper more and more situations will be encountered that require an ad hoc hypothesis to patch it up. Conversely, in the case of special relativity the expectation is that its small set of postulates is exhaustive; so far no ad hoc hypothesis has been needed, and that status is expected to remain so.

This economy is why special relativity has become the dominant theory.

Cleonis

In answer to my remark about the words "SR-spacetime" and "the aether" being interchageable you say "there is a difference" and then go on to remind us that LET and SR are indistinguishable. So how are the words different if the concepts they describe lead to the same outcome.

With regards to economy, I would regard it as perverse to choose the most difficult option if the choice makes no material difference.

Regarding acceleration, I can only assume that the theories agree on anything they may have to say about that otherwise the theories would be distinguishable and we would not have the luxury of choice as economy is no excuse for incorrect science.

Matheinste

 

[PeterDonis]

In all there is the two-step cascade:
- Velocity is time derivative of position
- Acceleration is time derivative of velocity.

We have that 4-position is not position with respect to spacetime; as a matter of principle an absolute reference of position does not enter special relativity.

It's true that there no invariant "position vector" for a point in spacetime in SR, but that doesn't mean that there is no invariant notion of velocity or acceleration; 4-velocity (defined as the derivative of position with respect to *proper time* along a worldline--i.e., the tangent vector to the worldline) and 4-acceleration (defined as the derivative of velocity with respect to *proper time* along a worldline--i.e., the rate of change of the tangent vector along the worldline) are both invariants, even though position itself is not. I'm not trying to argue for ether theory myself; I'm just pointing out that an ether theorist *could* use the invariant notions of 4-velocity and 4-acceleration as "velocity relative to the ether" and "acceleration relative to the ether", without being logically contradictory. (They'd still be violating Occam's Razor, of course, by multiplying assumptions beyond necessity, as you point out.)

 

[Cleonis]

[...] you say "there is a difference" and then go on to remind us that LET and SR are indistinguishable. So how are the words different if the concepts they describe lead to the same outcome.

You misquote me: I wrote that experimentally LET and SR are indistinguishable.

Basically you're letting on that you don't want to discuss these matters. I will respect that.

Cleonis

 

[atyy]

Why in the first place is there an "absolute position" in Newton and none in special relativity. Aren't both constructed using affine spaces?

The introductory textbooks have it right. Both Newton and special relativity respect the Principle of Relativity. The difference is whether c is finite or infinite.

In Newton the 3-acceleration is absolute. In special relativity the 4-acceleration is absolute.

 

[matheinste]

You misquote me: I wrote that experimentally LET and SR are indistinguishable.

Basically you're letting on that you don't want to discuss these matters. I will respect that.

Cleonis

By distinguish between them I meant decide which one more closely represents how the universe works. In that sense how else would you distinguish between them except by experiment. So although I (accidentally) left the word experimentally out I do not think its omission changes the meaning.

If I did not wish to discuss I would stop doing so.

Matheinste.

 

[D H]

You might want to read up on http://en.wikipedia.org/wiki/Fermi%E2%80%93Walker_transport" .

Thanks! I assume that this is the orientation with respect to which a ring laser gyro, or anything else sensitive to the Sagnac effect, measures. (If not, let me know.) That answers one question.

How about star trackers? When I google "rotating frame general relativity" I get lots of hits that reference Mach's Principle. The concept of the "fixed stars" does indeed seem to carry over onto general relativity. Yet massive objects curve space, so the fixed stars are not quite fixed in GR. I guess that can be fixed by relegating the term "fixed stars" to regions where local space-time is basically flat. Is this correct?

 

[PeterDonis]

I assume that this is the orientation with respect to which a ring laser gyro, or anything else sensitive to the Sagnac effect, measures.

I believe this is correct, at least as long as the Sagnac effect measurement is carried out over a small enough piece of the spacetime--i.e., it's a "local" measurement.

When I google "rotating frame general relativity" I get lots of hits that reference Mach's Principle.

Yep. :-)

The concept of the "fixed stars" does indeed seem to carry over onto general relativity. Yet massive objects curve space, so the fixed stars are not quite fixed in GR. I guess that can be fixed by relegating the term "fixed stars" to regions where local space-time is basically flat. Is this correct?

I guess it depends on why you want to use the term "fixed stars". If you're trying to use the stars as reference points, for example as navigational markers, the main effect you need to account for is their proper motions; some stars move detectably every year. That's a separate issue from how their observed positions relate to spacetime curvature.

However, if you're trying to use the "fixed stars" to explain why objects feel inertial forces (which is how Mach's Principle gets into the act), then the key question, as I understand it, is whether the universe as a whole is open or closed. If the universe is closed (meaning that it has a finite, if extremely large, volume--technically, its spatial slices have the topology of a 3-sphere), then you can legitimately say that all inertial forces felt by objects in the universe are ultimately due to the gravity of the universe as a whole, which determines its overall curvature. Since there is no spatial "boundary" to the universe, not even at spatial infinity (since there is no such thing in a closed universe), there are no "boundary conditions" to be specified over and above the solution of the Einstein field equation for the universe as a whole--which is just solving for how the gravity of the universe as a whole determines its curvature.

If the universe is open, however, then in addition to any solution we come up with for the EFE, we have to specify boundary conditions: specifically, what happens as we go to spatial infinity. So you can no longer claim, as you can with a closed universe, that all inertial forces felt by objects in the universe are solely due to its gravity; the extra boundary condition at spatial infinity also has an effect.

 

[D H]

I guess it depends on why you want to use the term "fixed stars". If you're trying to use the stars as reference points, for example as navigational markers, the main effect you need to account for is their proper motions; some stars move detectably every year.

There's a standard list of 57 stars used by star trackers chosen for brightness and small proper motion. By star tracker, I am talking about one of these (examples): http://www.ballaerospace.com/page.jsp?page=104" .

Ideally one would use quasars rather than visible stars, and that is exactly what is now used to define the best scientific guess regarding what constitutes a non-rotating frame. http://www.iers.org/products/2/12970/orig/message_150.txt" .

 

[Cleonis]

It's true that there no invariant "position vector" for a point in spacetime in SR, but that doesn't mean that there is no invariant notion of velocity or acceleration; 4-velocity (defined as the derivative of position with respect to *proper time* along a worldline--i.e., the tangent vector to the worldline) and 4-acceleration (defined as the derivative of velocity with respect to *proper time* along a worldline--i.e., the rate of change of the tangent vector along the worldline) are both invariants, even though position itself is not. I'm not trying to argue for ether theory myself; I'm just pointing out that an ether theorist *could* use the invariant notions of 4-velocity and 4-acceleration as "velocity relative to the ether" and "acceleration relative to the ether", without being logically contradictory.

Hmm, you raise an interesting point: if an ether theorist and a relativist each offer their best arguments, what will their discussion be like?

There are in a sense two relativity principles:
- The principle of relativity of position
- The principle of relativity of inertial motion.

In the case of, say, a sailing ship on a featureless sea, the sea is uniform and therefore has no intrinsic position reference. For the sailors there is no such thing as detecting where they are, but they can detect and log their velocity with respect to the sea. Two ships can part ways and using dead reckoning they can plot a course back to a rendez-vous point. (This rendez-vous point can for example be the pointship would be if it would have remained motionless.)

The principle of relativity of inertial motion is about regarding the continuum of all velocities as a uniform space in such a way that there is no intrinsic velocity reference. For spaceship crews there is no such thing as detecting where they are or what velocity they have, but they can detect and log their acceleration with respect to SR-spacetime. Two spaceships can part ways and using the acceleration counterpart of dead reckoning (applying the Minkowsk metric), they can plot a course back to a rendez-vous point. (This rendez-vous point can for example be the point where a spaceship would be if it would have remained in inertial motion.)


The relativist will argue that while an invariant 4-velocity can be defined, it is a mathematical artefact; no physical significance is attributed to it. That is, to say that special relativity has no concept of velocity relative to SR-spacetime is not saying that no definition of an invariant 4-velocity can be constructed, it's saying that SR disallows attributing physical significance to it.

In special relativity it is simply assumed that its set of fundamental principles will not give rise to self-contradiction, and so far none has been pointed out.

That is why I regard as the defining difference between special relativity and Lorentz ether theory: In terms of SR velocity relative to spacetime does not enter the theory, whereas LET does attribute physical significance to velocity relative to the ether.

Cleonis

 

[PeterDonis]

Cleonis: Why do you think SR attributes no physical significance to 4-velocity? It attributes no absolute significance to the *components* of 4-velocity, but the 4-velocity vector itself is another matter. 4-velocity is 4-momentum divided by rest mass, and 4-momentum certainly has physical significance, since it's the subject of a conservation law. As a relativist, I wouldn't want to take on the task of defending the proposition that 4-velocity itself has no physical significance but 4-momentum does.

Personally, I'm satisfied with the fact that ether theory violates Occam's Razor: it postulates something (the ether) that makes no difference to the results of any experiment, since we have a theory (SR) that predicts the same results for all experiments without postulating an ether.

 

[Cleonis]

Cleonis: Why do you think SR attributes no physical significance to 4-velocity? It attributes no absolute significance to the *components* of 4-velocity, but the 4-velocity vector itself is another matter. 4-velocity is 4-momentum divided by rest mass, and 4-momentum certainly has physical significance, since it's the subject of a conservation law. As a relativist, I wouldn't want to take on the task of defending the proposition that 4-velocity itself has no physical significance but 4-momentum does.

There seems to be some babylonian confusion going on here.

In Minkowski spacetime the conservation law is compliant with the principle of relativity of inertial motion.

I'm thinking now about the case of inelastic collision, such as collision experiments in particle accelerators. The relativistic point of view is that the amount of kinetic energy that is released comes from the relative velocity of the two objects that are involved in the collision. Nothing of what happens in that collision process is attributed to velocity with respect to SR-spacetime. (As we know, ether theories do attribute physical effects to velocity with respect to the assumed ether, with these physical effects being undetectable because the physical laws "conspire" against that.)

Of course, to create the ability to calculate the relative velocity, each of the velocities must be mapped in some coordinate system. In calculations assigning velocity is a necessary step, but that doesn't change the fact that no effect is attributed to velocity with respect to SR-spacetime.

It's not clear to me whether looking at the components of a 4-velocity vector or looking at "the 4-velocity vector itself" is a helpful distinction. What is "the 4-velocity vector itself"? It's the components that you're working with.


Another example: Bell's spaceship paradox.
- If two spaceships in formation are not accelerating then we have that there are no detectable physical effects (as measured for interactions between the two spaceships), and according to SR there are no physical effects in the first place.
- If two spaceships in formation are accelerating (in the direction of the line that connects them), each accelerating with exactly the same G-count, then the crews onboard those ships will detect physical effects: the distance between the ships will become larger, and their clocks, synchronized prior to the acceleration run, will go out of sync. These physical effects are attributed to the fact that the formation of spaceships is in acceleration with respect to SR-spacetime, and in that case the physical properties of SR-spacetime kick in.

SR has in common with ether theories that a background structure is implied, a background structure that participates in the physics taking place. The difference is whether any physical effects are attributed to velocity with respect to the background structure.

Cleonis

 

[Dale]

It's not clear to me whether looking at the components of a 4-velocity vector or looking at "the 4-velocity vector itself" is a helpful distinction. What is "the 4-velocity vector itself"? It's the components that you're working with.

Most definitely not! The vector itself is a geometric object, a member of a set known as a vector space which has several geometric properties such as an inner product which obeys certain rules. The components are the inner products of the vector with a particular set of other vectors called a basis. The vector is most definitely not the same as its components.

 

[PeterDonis]

Cleonis: Remember that I'm playing a sort of devil's advocate here, trying to imagine how an ether theorist (which I am not) would respond to the arguments you're making. (Although I'm trying not to say things that aren't actually true--just pointing out an ether theorist's possible alternate interpretation.)

In Minkowski spacetime the conservation law is compliant with the principle of relativity of inertial motion.

Yes, of course. No argument here.

It's not clear to me whether looking at the components of a 4-velocity vector or looking at "the 4-velocity vector itself" is a helpful distinction. What is "the 4-velocity vector itself"? It's the components that you're working with.

The 4-velocity itself is an invariant geometric object; it's the tangent vector to a given worldline at a given event. The components of the 4-velocity are the projection of that invariant geometric object into a given reference frame. The invariant geometric object could be interpreted as "velocity with respect to spacetime" because spacetime itself is the geometric structure within which the geometric object, 4-velocity, "lives".

(As a relativist, I agree that calling the 4-velocity "velocity with respect to spacetime" adds absolutely nothing to our ability to predict anything. But that doesn't mean the 4-velocity isn't an invariant geometric object.)

Another example: Bell's spaceship paradox.

Yes, the accelerating spaceships have 4-velocity vectors that are changing from event to event along their worldlines (with respect to their own proper time). Since the 4-acceleration is the rate of change of an invariant (the 4-velocity) with respect to an invariant (the proper time), it's no surprise that it's also an invariant.

If two spaceships in formation are not accelerating then we have that there are no detectable physical effects (as measured for interactions between the two spaceships), and according to SR there are no physical effects in the first place.

Not in the common rest frame of the two ships, no. But an observer moving relative to the ships will observe them to be Lorentz-contracted, and if that observer is able to measure stresses within the ships, he will measure the Lorentz contraction to be causing detectable compressive stress. (This can be seen by Lorentz-transforming the stress-energy tensor from the ships' rest frame into the moving frame.)

(Of course, as a relativist, I would pounce on this as evidence that it *is*, in fact, *relative* velocity, not "velocity relative to the ether", that has physical effects. But it does illustrate that you can't make a blanket claim that "relative velocity has no physical effects". It does. I know that in the case I just quoted, the ships have no relative velocity--but in the next case, they will.)

These physical effects are attributed to the fact that the formation of spaceships is in acceleration with respect to SR-spacetime, and in that case the physical properties of SR-spacetime kick in.

Well, the fact that

...the amount of kinetic energy that is released comes from the relative velocity of the two objects that are involved in the collision.

could equally well be due to the "physical properties of spacetime", namely those properties that require that energy and momentum are conserved. Also, the ships start out at rest relative to one another, but they don't stay that way, in either of their own rest frames. So are the effects they observe due to the acceleration itself, or just due to the fact that the acceleration changes their relative velocities so they're no longer at rest relative to one another?

(Again, as a relativist I would point out that none of this changes the fact that the accelerating case is very different from the case of inertial motion, and that the difference is fundamentally due to the fact that the accelerating observers *feel* an acceleration. I just don't know for sure that this would stop the ether theorist from trying to come up with a notion of "velocity with respect to spacetime" as well.)

 

[atyy]

Cleonis: Remember that I'm playing a sort of devil's advocate here, trying to imagine how an ether theorist (which I am not) would respond to the arguments you're making. (Although I'm trying not to say things that aren't actually true--just pointing out an ether theorist's possible alternate interpretation.)

Since you seem to be a relativist, may I assume that you are not an ether theorist, but an infinite number of ether theorists - one for each of the infinite number of inertial frames, each of which is as good as absolute space?

 

[Cleonis]

Cleonis: Remember that I'm playing a sort of devil's advocate here, trying to imagine how an ether theorist (which I am not) would respond to the arguments you're making. (Although I'm trying not to say things that aren't actually true--just pointing out an ether theorist's possible alternate interpretation.)

Let me try to clarify: my intention is to make an inventory of the relativistic point of view. I think your effort to sort of play the devil's advocate is worthwile.

What a relativist argues is ever so compelling to that relativist, but it will not be compelling to an ether theorist. Presumably that is your message, and I quite agree with that. Another way of saying the same: neither the relativistic interpretation nor ether theory interpretation are enforced by the observations: it's a judgement call.

Here is how I would argue if I would be etherially inclined:
Many introductions to SR suggest that SR is a relational theory. That is, introductions tend to emphasize only 'the principle of relativity' and that 'SR has done away with the notion of ether'. That carries a suggestion that space is just empty nothingness, and novices follow up on that suggestion. Novices who ask questions on physicsforums ask in near desperation: "But how can the twins be different of age when the traveller returns, if space is just empty nothingness?"

Another example: introductions to SR tend to emphasize things like relativistic doppler shift. As we know, relativistic doppler shift is purely a function of the relative velocity of emitter and detecter.

Now, SR is not a relational theory. By implication SR uses a background structure that participates in the physics taking place. Awkwardly, there is no specific name for the SR background structure, which really hampers communication. Shall we use the expression 'Minkowski spacetime' to refer to the background structure? Well, some people will insist that 'Minkowski spacetime' should be used only to refer to the mathematical concept, without direct physical interpretation. So we have that for an essential element of SR, its background structure, there is no identifying name!

Doing the splits

In effect introductions to SR are doing the splits. The novice is seduced into thinking that SR is a relational theory, but at the same time the introductions are implicitly providing the evidence that SR cannot be a relational theory, all without explanation. That raises the question: are authors of SR introductions even aware that SR isn't a relational theory?

That is how I would argue if I would play the devil's advocate.

Cleonis

 

[matheinste]

Cleonis,

----Novices who ask questions on physicsforums ask in near desperation: "But how can the twins be different of age when the traveller returns, if space is just empty nothingness?" ------

I have never heard anyone ask this, in desperation or otherwise.

Matheinste.

 

[Cleonis]

[...] an observer moving relative to the ships will observe them to be Lorentz-contracted, and if that observer is able to measure stresses within the ships, he will measure the Lorentz contraction to be causing detectable compressive stress. (This can be seen by Lorentz-transforming the stress-energy tensor from the ships' rest frame into the moving frame.)

I think it's necessary to retain strict distinction between actually measuring and 'inference on theoretical grounds'.

For example: the only way to actually measure acceleration (with respect to spacetime) is to use an actual accelerometer, onboard the accelerating spaceship.

In that sense there is no such thing as 'observing the acceleration in another frame of reference'. One can use a theory of physics to transfrom the actually measured acceleration to the acceleration as mapped in another coordinate system. Conversely, one can receive a radio signals from an accelerating spaceship and then one can use a theory of physics to infer what the acceleration as measured by onboard accelerometers must be.

Likewise, it seems to me that material stress is to be measured by a co-moving device.

Cleonis

 

[atyy]

I think it's necessary to retain strict distinction between actually measuring and 'inference on theoretical grounds'.

For example: the only way to actually measure acceleration (with respect to spacetime) is to use an actual accelerometer, onboard the accelerating spaceship.

In that sense there is no such thing as 'observing the acceleration in another frame of reference'. One can use a theory of physics to transfrom the actually measured acceleration to the acceleration as mapped in another coordinate system. Conversely, one can receive a radio signals from an accelerating spaceship and then one can use a theory of physics to infer what the acceleration as measured by onboard accelerometers must be.

Likewise, it seems to me that material stress is to be measured by a co-moving device.

Cleonis

You can't even define an accelerometer without a theory of physics. I give you an "accelerometer", and you accelerate in a Ferrari, yet the "accelerometer" reads zero. Are you going to conclude that the acceleration was zero?

 

[Cleonis]

You can't even define an accelerometer without a theory of physics.

I agree with that to the following extent:
Any entities of a theory must be defined operationally, and the operational definition is subject to the context that the theory provides.

Assumptions:
- The only long range forces besides gravitation is electromagnetic interaction. If we eliminate electromagnetic interaction we obtain a pure gravitational reading.
- The accelerometer consists of a chamber, inside that chamber an object is released to free motion. Then over a series of intervals of time the position of the object relative to the walls of the chamber is measured. (For example, the position of the object can be tracked with Doppler radar measurement.)
- The acceleration of the object relative to the chamber is the acceleration reading.

Comments:
Of course, tracking the object inside the chamber involves quite a bit of technology. Arguably the position of the object inside the chamber is measured indirectly, and its motion is inferred from the indirect measurements. To define acceleration a standard of length must be defined (so-and-so many wavelengths of a particular very reproducible emission line), a standard of time must be defined (so-and-so many oscillations of a particular very reproducible frequency.)

Many different setups can be used to measure acceleration, with various degrees of (in)directness. The setup with a tracked object that is released to free motion is the most direct setup, I think. Other accelerometer designs are calibrated against that.

Cleonis

 

[PeterDonis]

Here is how I would argue if I would be etherially inclined:

Switching sides, are you? :-)

Now, SR is not a relational theory. By implication SR uses a background structure that participates in the physics taking place. Awkwardly, there is no specific name for the SR background structure, which really hampers communication. Shall we use the expression 'Minkowski spacetime' to refer to the background structure? Well, some people will insist that 'Minkowski spacetime' should be used only to refer to the mathematical concept, without direct physical interpretation. So we have that for an essential element of SR, its background structure, there is no identifying name!

To me, "Minkowski spacetime" is fine as a name for the SR background structure. I haven't seen anyone object that the term should only be used to refer to the mathematical concept; I have seen people object that Minkowski spacetime is physically unrealistic because it's globally flat, whereas no real physical spacetime would be exactly globally flat (in space *and* time--of course there are many spacetimes that have globally flat spatial slices, but that's not the same thing). You're quite correct that Minkowski spacetime *is* a background structure in SR; it is not changed by any dynamics of the system under consideration, unlike in GR.

The novice is seduced into thinking that SR is a relational theory, but at the same time the introductions are implicitly providing the evidence that SR cannot be a relational theory, all without explanation. That raises the question: are authors of SR introductions even aware that SR isn't a relational theory?

Can you give any specific examples of introductions that you think are doing this? I learned SR from Taylor and Wheeler's _Spacetime Physics_, and the main principle I took away from that is that the theory of "relativity" is actually about *invariants*--things that *don't* change when you change reference frames. IIRC, Taylor even makes an explicit statement somewhere in the book that "relativity" is a bad name, and the theory really should be called the "theory of invariants". I think there's also a similar statement in the classic GR text, Misner, Thorne, and Wheeler's _Gravitation_.

 

[atyy]

I agree with that to the following extent:
Any entities of a theory must be defined operationally, and the operational definition is subject to the context that the theory provides.

Assumptions:
- The only long range forces besides gravitation is electromagnetic interaction. If we eliminate electromagnetic interaction we obtain a pure gravitational reading.
- The accelerometer consists of a chamber, inside that chamber an object is released to free motion. Then over a series of intervals of time the position of the object relative to the walls of the chamber is measured. (For example, the position of the object can be tracked with Doppler radar measurement.)
- The acceleration of the object relative to the chamber is the acceleration reading.

Comments:
Of course, tracking the object inside the chamber involves quite a bit of technology. Arguably the position of the object inside the chamber is measured indirectly, and its motion is inferred from the indirect measurements. To define acceleration a standard of length must be defined (so-and-so many wavelengths of a particular very reproducible emission line), a standard of time must be defined (so-and-so many oscillations of a particular very reproducible frequency.)

Many different setups can be used to measure acceleration, with various degrees of (in)directness. The setup with a tracked object that is released to free motion is the most direct setup, I think. Other accelerometer designs are calibrated against that.

Cleonis

How do you define "free motion"?

 

[Cleonis]

How do you define "free motion"?

That was already covered in the assumptions declared at the start:
Assumption:
- The only long range force besides gravitation is electromagnetic interaction. If we eliminate electromagnetic interaction we obtain a pure gravitational reading.

Cleonis

 

[atyy]

That was already covered in the assumptions declared at the start:
Assumption:
- The only long range force besides gravitation is electromagnetic interaction. If we eliminate electromagnetic interaction we obtain a pure gravitational reading.

Cleonis

How do you know you've eliminated the electromagnetic interaction?

If we're discussing special relativity, then the "gravitational reading" means flat spacetime.

I don't think you can define any of these terms until you already know about inertial frames. When you do know about inertial frames, you also know about noninertial frames. So there is no difference between "actual measurement" and "inference on theoretical grounds".

 

[Cleonis]

Switching sides, are you? :-)
I learned SR from Taylor and Wheeler's _Spacetime Physics_, and the main principle I took away from that is that the theory of "relativity" is actually about *invariants*--things that *don't* change when you change reference frames. IIRC, Taylor even makes an explicit statement somewhere in the book that "relativity" is a bad name, and the theory really should be called the "theory of invariants". I think there's also a similar statement in the classic GR text, Misner, Thorne, and Wheeler's _Gravitation_.

Certainly, emphasizing the importance of thinking in invariants over thinking in terms of relativity is better.
Then again: the feature of invariants is not unique to SR, galilean relativity has its own invariants.

In my opinion any text that claims to be an introduction to SR ought to emphasize precisely the feature that differentiates SR from galilean relativity. SR-spacetime affects how objects relate to each other in space and in time. SR-spacetime is an active participant in the physics taking place.

Compared to SR-spacetime galilean spacetime is pretty passive. In galilean spacetime, when two twins separate and later rejoin then the stay-at-home twin and the traveling twin won't notice anything special. In SR-spacetime however, the twins find that the difference in their journeys has had a physical effect.

The transition from galilean relativity to SR was a transition from a comparitively passive spacetime to a spacetime that is an active participant in the physics taking place. In my opinion in any introduction to SR the story ought to revolve around that.

Cleonis

 

[PeterDonis]

The transition from galilean relativity to SR was a transition from a comparitively passive spacetime to a spacetime that is an active participant in the physics taking place. In my opinion in any introduction to SR the story ought to revolve around that.

I suspect that most relativists would say that GR is the theory that has spacetime being an active participant in the physics. Spacetime in SR is predetermined--as you know, since you've called it a "background structure". It's always flat Minkowski spacetime. Only in GR is spacetime affected by the dynamics.

In fact, if I wanted to quickly summarize the difference between galilean relativity and SR, I would simply say that galilean relativity uses galilean spacetime, whereas SR uses Minkowski spacetime. The physical effects you cite (e.g., the twins aging differently) I would not say are due to spacetime "participating" more in SR--they're just due to SR using a different spacetime. Galilean spacetime, if we lived in it, would have physical effects too: after all, objects are predicted to experience inertial forces (e.g., centrifugal force) in pre-SR physics.

 

[Cleonis]

if I wanted to quickly summarize the difference between galilean relativity and SR, I would simply say that galilean relativity uses galilean spacetime, whereas SR uses Minkowski spacetime. [...] Galilean spacetime, if we lived in it, would have physical effects too: after all, objects are predicted to experience inertial forces (e.g., centrifugal force) in pre-SR physics.

The theme of inertia is worth a thread of its own, I think. I will start a new thread, copying this quote.
The theme is: inertia in theories of motion.

I will call the new thread: 'History of theories of motion; the role of inertia'.

Cleonis