# Acceleration and velocity: newtonian versus relativistic interpretation.

1. Sep 12, 2009

### Cleonis

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

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

2. Sep 12, 2009

### atyy

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.

3. Sep 12, 2009

### matheinste

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

Matheinste.

4. Sep 12, 2009

### atyy

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

5. Sep 12, 2009

### JesseM

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.
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.
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.

6. Sep 13, 2009

### Staff: Mentor

By the definition given in the http://en.wikipedia.org/wiki/Background_independence" [Broken], 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.

Last edited by a moderator: May 4, 2017
7. Sep 13, 2009

### Staff: Mentor

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.

8. Sep 13, 2009

### JesseM

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.

9. Sep 13, 2009

### D H

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

10. Sep 13, 2009

### JesseM

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 [Broken]), 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.

Last edited by a moderator: May 4, 2017
11. Sep 13, 2009

### D H

Staff Emeritus
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.

12. Sep 13, 2009

### JesseM

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 [Broken] 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.

Last edited by a moderator: May 4, 2017
13. Sep 13, 2009

### Cleonis

No problem, I don't venture into metaphysics.

(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

14. Sep 13, 2009

### matheinste

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.

15. Sep 13, 2009

### Staff: Mentor

Hi Cleonis,

16. Sep 13, 2009

### Cleonis

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

17. Sep 13, 2009

### Cleonis

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

18. Sep 13, 2009

### Staff: Mentor

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.

19. Sep 13, 2009

### 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"?

20. Sep 13, 2009

### atyy

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.