Acceleration and velocity: Newtonian versus relativistic interpretation.

[Cleonis]

In another thread a discussion arose about the interpretation of derivatives. For example acceleration is the time derivative of velocity.

[...] coordinate velocity is the derivative of coordinate position, and coordinate acceleration is the derivative of coordinate velocity (and proper acceleration at a given point on an object's worldline is just the coordinate acceleration in the object's instantaneous inertial rest frame at that point, which will match the reading on a co-moving accelerometer at that point). Do you agree that in the mathematical sense this is as true in relativity as it is in classical physics? If you do agree, are you saying that even though it's true mathematically, it's not true "in physical interpretation" in relativity? [...]

Obviously, in the strictly mathematical sense: in relativistic physics and classical physics alike acceleration is the time derivative of velocity.

Let me first discuss an example where both in the mathematical sense and in the sense of physical interpretation there is a process of derivation.

Let there be two coils of conducting wire, I will refer to them as the 'primary coil' and the 'secondary coil'.
Electrically charged particles (electrons) are located in the conductors. As we know, the electrical counterpart of velocity is current. Current is the first time derivative of charge position. The electrical counterpart of acceleration is change of current strength.
If there is change of current strength in the primary coil a current is induced in the secondary coil.

Now, what can you infer when you observe:
- the presence of the primary coil (but you cannot directly observe whether the primary coil even consists of conducting wire.)
- a sinusoidal alternating current in the secondary coil.

Observing what current is induced in the secondary coil is highly informative; it allows you to reconstruct with high fidelity what the current is in the primary coil.
In turn, inferring the existence of current in the primary is immediate proof that the primary consists of a conducting wire.

My point is: the reason you can make those inferences is that change in current strength is physically a derivative process, and the mathematical operation reflects a physical dependency. If a change of current strength exists then a current must exist also, and if there is current then free-to-flow electric charge must be present in the first place.

Or take the example of a single coil, with self-induction. Then there will be inductance at play. When you apply an electromotive force a current will tend to start, but immediately the self-induction kicks in. The change in current strength induces a changing electromagnetic field, which opposes the change in current strength. Inductance is analogous to inertia; change in current strength is proportional to the applied electromotive force.

There is a Newtonian interpretation of dynamics that can be seen as a one-on-one analogy with inductance. According to this interpretation:
'If objects accelerate with respect to the absolute space then velocity with respect to the absolute space must exist also, and if there is velocity relative to the absolute space then absolute position with respect to the absolute space must exist also.'

Relativistic theories of motion affirm the existence of acceleration with respect to the background structure, but as a matter of principle velocity with respect to the background structure does not exist in a relativistic theory.

(Many names for the background structure are in circulation: some refer to it as 'Minkowski spacetime', some refer to it as 'an inertial frame of reference', some refer to it as 'inertial space', etc. Since a background structure exists anyway the most neutral name seems just that: 'the background structure'.)

As a mathematical operation, acceleration is defined as the time derivative of velocity, but in relativistic theories this does not reflect a physical dependency.

Cleonis

 

[atyy]

What I learned from Arthur Beiser instead of your fancy philosophers:

There is no preferred frame - there are many of them - there is a class of preferred frames - inertial frames. This is true in Newton and special relativity. So there is no absolute space in Newton anyway.

 

[matheinste]

Relativistic theories of motion affirm the existence of acceleration with respect to the background structure, but as a matter of principle velocity with respect to the background structure does not exist in a relativistic theory.

(Many names for the background structure are in circulation: some refer to it as 'Minkowski spacetime', some refer to it as 'an inertial frame of reference', some refer to it as 'inertial space', etc. Since a background structure exists anyway the most neutral name seems just that: 'the background structure'.)

Cleonis

Perhaps I am getting confused by the terminology but isn't GR a background independent theory.

Matheinste.

 

[atyy]

Perhaps I am getting confused by the terminology but isn't GR a background independent theory.

Matheinste.

Yes, and it has absolute 4-velocity too (the tangent vector to a particle's wordline), doesn't it?

 

[JesseM]

There is a Newtonian interpretation of dynamics that can be seen as a one-on-one analogy with inductance. According to this interpretation:
'If objects accelerate with respect to the absolute space then velocity with respect to the absolute space must exist also, and if there is velocity relative to the absolute space then absolute position with respect to the absolute space must exist also.'

This sounds like metaphysics, not physics. Even in Newtonian physics, if the laws of physics are Galilei-invariant then there is no way to determine experimentally which frame is at rest in absolute space. If you want to ground your definitions in experiments involving physical apparatuses, you must define position, velocity and acceleration relative to an inertial coordinate system constructed out of rulers and clocks, just as in relativity.

Relativistic theories of motion affirm the existence of acceleration with respect to the background structure

Do they? Again, all measurable definitions of position, velocity, and acceleration are defined relative to coordinate grids in relativity just as in Newtonian physics. If you're talking about something metaphysical, then you shouldn't use the phrase "relativistic theories of motion affirm...", you should just say something like "certain physicists and philosophers have a metaphysical interpretation of relativity which affirms..." or something along those lines. And I actually doubt most physicists would be willing to affirm this unless you defined what you meant more clearly.

(Many names for the background structure are in circulation: some refer to it as 'Minkowski spacetime', some refer to it as 'an inertial frame of reference', some refer to it as 'inertial space', etc. Since a background structure exists anyway the most neutral name seems just that: 'the background structure'.)

But "Minkowski spacetime" is a totally different notion than "an inertial frame of reference"--also, if you want to talk about inertial frames, an object's velocity relative to a given frame is every bit as well-defined as its acceleration relative to a given frame. The difference is that all inertial frames agree whether an object has zero or nonzero acceleration, but disagree on whether it has zero or nonzero velocity. "Minkowski spacetime" is probably a better candidate for what you're getting at, since non-accelerating objects have geodesic worldlines through Minkowski spacetime (they maximize the proper time between points, the proper time along any worldline being determined by the Minkowski metric) while accelerating objects don't follow geodesics.

 

[PeterDonis]

Perhaps I am getting confused by the terminology but isn't GR a background independent theory.

Matheinste.

By the definition given in the http://en.wikipedia.org/wiki/Background_independence" , yes, it is, but there are a number of theoretical issues involved. Here's the definition:

Background independence is a condition in theoretical physics, especially in quantum gravity (QG), that requires the defining equations of a theory to be independent of the actual shape of the spacetime and the value of various fields within the spacetime, and in particular to not refer to a specific coordinate system or metric.

The standard formulation of GR meets that requirement; however, as the discussion in the Wikipedia page shows, there are plenty of subtleties to the issue.

 

[PeterDonis]

"Minkowski spacetime" is probably a better candidate for what you're getting at, since non-accelerating objects have geodesic worldlines through Minkowski spacetime (they maximize the proper time between points, the proper time along any worldline being determined by the Minkowski metric) while accelerating objects don't follow geodesics.

Actually, this is true in any spacetime in GR, not just Minkowski spacetime. In fact, it's a defining property of geodesics that they are unaccelerated worldlines (i.e., observers following them feel no acceleration). So there is a coordinate-independent definition of "acceleration" that doesn't require it to be "the time derivative of velocity": "acceleration" simply means "non-geodesic motion". Since geodesics are coordinate-independent objects, and don't require any notion of "velocity" (or even "position") for their definition, this works just fine.

 

[JesseM]

Actually, this is true in any spacetime in GR, not just Minkowski spacetime.

I understand that, but I thought Cleonis was just talking about the difference between Newtonian physics and SR in this thread, without getting into issues related to curved spacetime.

 

[D H]

The difference is that all inertial frames agree whether an object has zero or nonzero acceleration, but disagree on whether it has zero or nonzero velocity.

Doesn't this imply some kind of background in GR? Otherwise, how do you explain how accelerometers and ring laser gyros work?

 

[JesseM]

Doesn't this imply some kind of background in GR? Otherwise, how do you explain how accelerometers and ring laser gyros work?

I don't exactly understand the question--what do you mean by "background", and why do you ask about GR specifically? As PeterDonis said, defining acceleration in terms of a non-geodesic path works in GR just as it does in SR, and an accelerometer on a non-geodesic path will always measure a nonzero reading (while a geodesic path represents a freely-falling object in GR, who will always feel weightless so their accelerometer reads zero). Also, in GR we have the concept of a locally inertial frame in an infinitesimal region of spacetime (see the http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html ), so I assume a geodesic path would be moving at constant velocity in such a locally inertial frame while a non-geodesic path would not.

 

[D H]

I quoted the wrong text, and I am apparently a bit off in my interpretation of "background independent". What I was getting at with accelerometers is that there is what I would call a local background, aka a geodesic, against which accelerometers measure acceleration. Ring laser gyros measure something too against a background of some sort, and its not just acceleration.

 

[JesseM]

I quoted the wrong text, and I am apparently a bit off in my interpretation of "background independent". What I was getting at with accelerometers is that there is what I would call a local background, aka a geodesic, against which accelerometers measure acceleration. Ring laser gyros measure something too against a background of some sort, and its not just acceleration.

A geodesic is a type of worldline in spacetime (akin to a line or curve drawn on a piece of paper), so I don't understand what you mean when you equate a geodesic with a "local background". And "background independence" seems to be sort of a subtle idea which I'm not sure I understand, http://www.aei.mpg.de/einsteinOnline/en/spotlights/background_independence/index.html says it's closely related to the notion of "diffeomorphism invariance" which basically means the equations for the laws of physics will look the same in any arbitrary coordinate system, but I don't quite follow what the difference between background independence and diffeomorphism invariance is supposed to be (according to the page it's something to do with formulating the laws of physics in terms of 'relations' between physical entities and events rather than in terms of abstract background coordinate systems, but that's a little vague)

Also, I do think a ring laser gyro measures a type of acceleration--"changes in orientation and spin" according to the wikipedia intro. If you have a ring that's changing orientation or spin, any given point on that ring should be moving on a non-geodesic worldline, I would think.

 

[Cleonis]

If you're talking about something metaphysical, then you shouldn't use the phrase "relativistic theories of motion affirm...", [...]

No problem, I don't venture into metaphysics.

[...] The difference is that all inertial frames agree whether an object has zero or nonzero acceleration, but disagree on whether it has zero or nonzero velocity.

(The context of this posting is Minkowski spacetime. I didn't extend the discussion to GR spacetime because that would make the posting too bulky.)

When discussing acceleration it must always be stated with respect to what that acceleration is. (Of course, when it is clear from the context then it can be omitted.)

In this case, let me state the reference of the acceleration explicitly. The set of all inertial frames of reference constitutes the equivalence class of inertial frames of reference. In special relativity acceleration is with respect to the equivalence class of inertial frames of reference.

Special relativity affirms uniqueness of acceleration; when an object is accelerating then the G-count of that acceleration is frame-independent.

Another consideration, when questions are submitted about the twin scenario, the answer is always (and correctly): there is no full symmetry between the stay-at-home twin and the traveller. The worldline of the traveller is not straight, and the only way to have a non-straight worldline is to be subject to acceleration.
Special relativity affirms the existence of acceleration with respect to the equivalence class of inertial frames of reference, in the sense that the acceleration is invoked to account for the non-symmetry of the stay-at-home worldline versus the traveler's worldline.

In my opinion there is nothing "metaphysical" about affirming the existence of acceleration. It's part of the regular, generally applied stuff of special relativity.

Cleonis

 

[matheinste]

No problem, I don't venture into metaphysics.



Special relativity affirms the existence of acceleration with respect to the equivalence class of inertial frames of reference, in the sense that the acceleration is invoked to account for the non-symmetry of the stay-at-home worldline versus the traveler's worldline.

In my opinion there is nothing "metaphysical" about affirming the existence of acceleration. It's part of the regular, generally applied stuff of special relativity.

Cleonis

Isn't that exactly what most of us have been saying. What you have just said is not the same thing as acceleration being relative to spacetime. As for the last comment I cannot see its relevance as no one is denying the existence of acceleration.

To PeterDonis,

Thanks for your comments on background dependency. I am aware that there is some debate concerning the matter, mostly, as yet and perhaps for always, way above my head.

Matheinste.

 

[Dale]

Hi Cleonis,

Sorry, it may be adult ADD kicking in or something, but I have read all of your comments in this thread and I have absolutely no idea what your point is. Can you please state it concisely.

 

[Cleonis]

Isn't that exactly what most of us have been saying. What you have just said is not the same thing as acceleration being relative to spacetime.

Indeed I should have stated it more sharply, as follows:
By implication special relativity affirms the existence of acceleration relative to spacetime.

Here, with spacetime, I mean the background structure. Motion in Minkowski spacetime has properties, these are properties of the spacetime. The differential aging in the twin scenario is a physical property of Minkowski spacetime.

I'm aware that some people advocate a more cautious attitude, advocating that we should regard the concept of Minkowski spacetime as a mental construct only, serving as a device to represent certain relations economically. I think that level of caution is overdoing it.

In physics, once a concept has proven to be efficient and economical in describing physics it is common to regard it as real.
For example, the concept of a field. We have no way of detecting an electromagnetic field directly, what we can observe is that electrically charged particles behave in certain ways. It's possible to formulate a theory of electromagnetic interaction in such a way that everyting is described in terms of potentials between particles (including velocity dependent potentials), but that formulation is cumbersome compared to a formulation in which an electromagnetic field exists that acts as mediator of electromagnetic interaction.

It seems to me the same applies for the concept of Minkowski spacetime. Do fields really exist? Does Minkowski spacetime really exist? I don't know, what I can say is that I put those in the same league in terms of the level of reality I attribute to them.

Cleonis

 

[Cleonis]

[...] I have read all of your comments in this thread and I have absolutely no idea what your point is. Can you please state it concisely.

Consicely:
As a matter of principle, the concept of velocity with respect to spacetime does not enter special relativity. By necessity, special relativity does affirm the existence of acceleration with respect to spacetime.

Mathematically, acceleration is derived from velocity.

Cleonis

 

[Dale]

Yes. Relativity assigns physical significance to the bending of a worldline and to its length (interval). Relativity does not assign physical significance to a worldline's direction. There is nothing in the former that contradicts the latter.

 

[DrGreg]

Physicists don't use the phrase "acceleration with respect to spacetime" because it's somewhat vague (and also potentially misleading in that it might encourage the reader to think that spacetime has some aethereal qualities that it doesn't).

They say instead "acceleration with respect to a co-moving inertial observer" because that is much more precise. And it's such an important concept they've given it a name, "proper acceleration". And it has a physical interpretation "acceleration as measured by an accelerometer".

When you talk specifically of spacetime rather than space, in a coordinate-independent way, then it's better to use geometrical language instead and talk about "curvature of a worldline".

When you draw a curved line on a flat piece of paper, do you say it is "curved with respect to the paper"?

 

[atyy]

When you draw a curved line on a flat piece of paper, do you say it is "curved with respect to the paper"?

How about curved with respect to a straight ruler? Spacetime is manifold and metric, and metric is the ruler, so if "curved wrt ruler" is ok, then "curved wrt spacetime" is ok too.

 

[DrGreg]

When you draw a curved line on a flat piece of paper, do you say it is "curved with respect to the paper"?

How about curved with respect to a straight ruler? Spacetime is manifold and metric, and metric is the ruler, so if "curved wrt ruler" is ok, then "curved wrt spacetime" is ok too.

The point is, you wouldn't say "curved with respect to the paper", you'd just say "curved", because curvature is a geometric invariant. Whereas the position of a line or the slope of a line are not invariants and have to be measured relative to something else. I'm making the analogy here between "curvature" and "proper acceleration".

 

[Dale]

and metric is the ruler

I wouldn't say that. The metric is the mathematical equation that relates the measurements of rulers and clocks to the coordinates and vice versa.

 

[atyy]

The point is, you wouldn't say "curved with respect to the paper", you'd just say "curved", because curvature is a geometric invariant. Whereas the position of a line or the slope of a line are not invariants and have to be measured relative to something else. I'm making the analogy here between "curvature" and "proper acceleration".

I see. I was thinking 4-acceleration.

 

[Cleonis]

Physicists don't use the phrase "acceleration with respect to spacetime" [...]

I am aware of the convention to say: "acceleration with respect to the instantaneously co-moving inertial observer."
Or the version: "acceleration with respect to the instantaneously co-moving inertial frame."

It seems to me that such a phrasing is potentially misleading in that it might encourage the reader to think that 'the observer', or 'the frame' have some particular physical significance, which they don't.

It seems to me invocation of 'the observer' or 'the frame' is superfluous, in the sense that omitting them doesn't render the physical representation incomplete. When an accelerometer registers acceleration we assume that some causal process is unvolved. An accelerometer registers acceleration when it is being accelerated with respect to spacetime.

(Of course I'm aware it would be pointless to deviate from the ingrained conventions without explanation. But I think it's interesting to examine the soundness of the ingrained conventions.)

When you talk specifically of spacetime rather than space, in a coordinate-independent way, then it's better to use geometrical language instead and talk about "curvature of a worldline".

When you draw a curved line on a flat piece of paper, do you say it is "curved with respect to the paper"?

Indeed, a line counts as curved only when it has curvature with respect to the intrinsic geometry of the manifold that it is residing in. Similarly, I can draw a euclidean 2-space onto a piece of paper, and when I curve the paper the euclidean 2-space represented on the paper remains that euclidean 2-space.

In the context of special relativity the spacetime is thought of as immutable and as having intrinsically flat geometry. To say "this object is accelerating with respect to spacetime" is intended as stating "this object is accelerating with respect to the intrinsic geometry of the spacetime."

Cleonis

 

[atyy]

(Of course I'm aware it would be pointless to deviate from the ingrained conventions without explanation. But I think it's interesting to examine the soundness of the ingrained conventions.)

Tangent space, manifold, worldline, curve - worldline is NOT a curve in some terminologies - a curve is a parameterized worldline. Tangent space is related to the instantaneously comoving inertial observer.

Where's Fredrick when we need him?

 

[atyy]

Indeed, a line counts as curved only when it has curvature with respect to the intrinsic geometry of the manifold that it is residing in. Similarly, I can draw a euclidean 2-space onto a piece of paper, and when I curve the paper the euclidean 2-space represented on the paper remains that euclidean 2-space.

You must mean "dissimilarly". A line has no intrinsic curvature, but dissimilarly, a 2D surface can have intrinsic curvature.

 

[atyy]

Doesn't this imply some kind of background in GR? Otherwise, how do you explain how accelerometers and ring laser gyros work?

I think as long as you use the formulation Einstein Equation plus geodesic equation of motion, then there is a background, because the mass of your test particle was not included as a cause of curvature when you solved the Einstein Equation.

There is a more general formulation in which you write the Einstein Equation plus the equations of state (like the Maxwell equations) of all matter - no test particles - no equation of motion for test particles. This is fully background independent (in the GR sense, there's still background like the signature of the metric). The geodesic equation of motion is recovered in the "ray limit" of the full "physical optics" formulation.

 

[PeterDonis]

Indeed, a line counts as curved only when it has curvature with respect to the intrinsic geometry of the manifold that it is residing in. Similarly, I can draw a euclidean 2-space onto a piece of paper, and when I curve the paper the euclidean 2-space represented on the paper remains that euclidean 2-space.

In the context of special relativity the spacetime is thought of as immutable and as having intrinsically flat geometry. To say "this object is accelerating with respect to spacetime" is intended as stating "this object is accelerating with respect to the intrinsic geometry of the spacetime."

This way of putting it is fine, and is equivalent to the way I was stating it. The "intrinsic geometry of the spacetime" is what determines which worldlines are geodesics and which are not. The geodesics are the "inertial" worldlines; the non-geodesics are the "accelerated" worldlines. This applies in *any* spacetime, not just the flat spacetime of special relativity.

The difference in SR, btw, is not that spacetime is "immutable"--just that it's flat. A "spacetime" in GR is an equally immutable object; it's just that its geometry need not be flat--it might be curved by the presence of mass-energy, and the curvature might vary from event to event in the spacetime, whereas in SR it's always zero--I suspect this is what you were getting at with the word "immutable". But a curved spacetime, considered as a single consistent solution of the Einstein field equation, is just as "immutable" as the flat spacetime of SR. The spacetime itself doesn't change as you "move through it", any more than the Earth's curved surface changes as you move through it.

I mention this point because which worldlines are geodesics, and which are not, is also "immutable" once you've specified a spacetime (in other words, once you've specified a particular solution of the Einstein field equation--the flat spacetime of SR is such a solution, btw, one in which the stress-energy tensor is zero at all events in the spacetime). So "acceleration", if we're careful to define it as "traveling along a non-geodesic worldline", is a perfectly good invariant (i.e., coordinate-independent) concept.

 

[PeterDonis]

Mathematically, acceleration is derived from velocity.

Given my previous post, it's easy to tackle this one. Mathematically, you're correct, but I don't think it means what you think it means. :-)

Here's the mathematical definition of "acceleration" which corresponds to the one I gave in my previous post (i.e., this is the mathematical definition of "the extent to which an observer is traveling along a non-geodesic worldline"):

"Acceleration" is the covariant derivative of the 4-velocity along a worldline, with respect to proper time along that worldline.

Or, in mathematical notation (from the http://en.wikipedia.org/wiki/Proper_acceleration" ):

A^{\lambda} = \frac{D U^{\lambda}}{d \tau}

which is, I assume, what you were thinking of when you said what I quoted above.

The http://en.wikipedia.org/wiki/Four-velocity 4-velocity U^{\lambda} at a given event on a worldline is simply the tangent 4-vector to the worldline at that event. In order to write it in a coordinate-independent manner, we parametrize the worldline by proper time, and use the covariant derivative:

U^{\lambda} = \frac{D x^{\lambda}}{d \tau}

Combining the two equations above, we have:

A^{\lambda} = \frac{D^2 x^{\lambda}}{d \tau^2}

This is written in terms of coordinate-independent quantities, and therefore the value calculated for A at a given event on a given worldline will be *the same in all reference frames*. So this should satisfy both your needs and mine: it shows how the notion of "derivative of velocity" gives rise to the acceleration, but it also shows how acceleration, so defined, is an invariant. (The 4-velocity, as defined above, btw, is also an invariant--tangent vectors to curves are just as intrinsic to a given geometry as the other things we've been discussing. So this should put to rest any qualms about how acceleration can be invariant while velocity is not. You just need to use the proper definition of velocity.)

To sum up: we define the invariant "4-velocity" as "the tangent vector to a worldline". We then define the invariant "acceleration" as "the rate at which the tangent vector changes along a worldline, with respect to proper time". A geodesic is simply a worldline whose invariant "acceleration" is zero all along itself; i.e., its tangent vector "doesn't change" in the invariant sense we've defined here. (The usual jargon is that a geodesic "parallel transports its tangent vector along itself"--the covariant derivative is simply the mathematical object that captures the invariant notion of "parallel transport".) A non-geodesic, or accelerated, worldline is one whose tangent vector "changes" from event to event.

 

[atyy]

Where's Fredrick when we need him?

To sum up: we define the invariant "4-velocity" as "the tangent vector to a worldline". We then define the invariant "acceleration" as "the rate at which the tangent vector changes along a worldline, with respect to proper time". A geodesic is simply a worldline whose invariant "acceleration" is zero all along itself; i.e., its tangent vector "doesn't change" in the invariant sense we've defined here. (The usual jargon is that a geodesic "parallel transports its tangent vector along itself"--the covariant derivative is simply the mathematical object that captures the invariant notion of "parallel transport".) A non-geodesic, or accelerated, worldline is one whose tangent vector "changes" from event to event.

This is close enough to Fredrick, I think ...