Acceleration and velocity: Newtonian versus relativistic interpretation.

[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