# Relativity of Acceleration

1. Apr 27, 2007

### fireball3004

A question has occurred to me that I am unable to answer. Consider a universe without other mater, besides that which is described. In this universe there is one person (you) and a chain (stretched along the x-axis) both accelerating around and aligned on the y-axis one above the other such that you do not view it as moving. The chain should from your point of view not change positions or experience any change what so ever, however I "know" that the chain expands do to inertia, experiences time dilation, and curves space time. All these phenomena are visible signs of movement but if you don't view the chain as moving... syntax error.
I have heard, though not read, Mach's point of view that acceleration is relative to all mater in the universe, but I am also aware that point of view is not widely accepted. So I am left with my question of "what is acceleration relative to?".

Last edited: Apr 27, 2007
2. Apr 27, 2007

### Mentz114

In order to define acceleration here must be some frame of reference. If a single body ( the only one ) accelerated by ejecting some matter, you would have to say that the acceleration is in the frame of the matter before it moved. In any case there are now two bodies so motion can be defined. But while the acceleration was happening there was only one, non-rigid body. If a body is changing shape I'm sure that's enough to give a frame.

Either way, if there's accelearation, there's a frame.

If this is wrong, I'm sure the relativity police will tell us.

Last edited: Apr 27, 2007
3. Apr 27, 2007

### MeJennifer

A perfectly born Rigid rod that accelerates in a completely empty universe would certainly give measurable effects. Each slice of the rod orthogonal to the direction of acceleration will experience a different temporal relationship with other slices. One could say that time is out of phase in the rod. We could force the temporal relationships between the slices in the rod to stay as before (think for instance of the Bell Spaceship paradox) but then we will observe that the spatial distances between the slices increase.

It is impossible to accelerate a rod and keeping both the temporal and spatial relationships between the slices.

Last edited: Apr 27, 2007
4. Apr 27, 2007

### Mentz114

This cannot happen, surely ?

5. Apr 27, 2007

### MeJennifer

I see no reason why not.

Similarly we can think of a rotating ball in a completely empty universe. Also in this case we can see a change in the temporal and spatial relationships.

A salient detail here is that while in the case of an accelerating rod we can fix either the spatial or temporal relationship but in the case of a rotating ball we cannot, both the spatial and temporal relationships are continiously changed.

6. Apr 27, 2007

### Mentz114

If it ejects part of itself, it can accelerate, in which case we have 2 frames.
If it is completely alone and it does not eject, what accelerates it ?

Can you back up your assertion with a reference or some equations ? I am eager to learn.

 I don't know exactly what you mean by spatial and temporal relationships. Please explain

Last edited: Apr 27, 2007
7. Apr 27, 2007

### Chris Hillman

Acceleration, anyone?

Well, you can't answer that question "in a theoretical vacuum". That is, you need to say something about what theory/model you have in mind! I'll assume the obvious default: anyone asking about spacetime is thinking of general relativity unless they specify otherwise. Then your question is easily answered: in relativity (special and general), the linear acceleration of a small object is identified with the path curvature of the world line of the object. (Path-curvature is a coordinate-free property of a parameterized curve, which makes perfect sense for curves in curved manifolds, but should not be confused with the curvature tensor of the spacetime itself. A geodesic is simply a curve with vanishing path curvature, which is why, in gtr, the world line of an unaccelerated test particle is always a timelike geodesic.)

Why did I say linear acceleration? Well, I'll let you think about how to formulate the angular momentum of an object spinning about an axis of rotation. (In the context of Mach principles, angular acceleration, in particular "Newton's bucket" thought experiment, is more often discussed in the literature than linear acceleration.)

Well, the first thing you need to know is that many distinct "Mach principles" have been formulated more or less precisely during the past century, some of which are less controversial (at least in some circumstances) than others. Here are a few suggestions to start your reading:

Try the book Space, TIme, and Space-time by Sklar cited at http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#phil [Broken] , which informally discusses both "Newton's bucket" and rotation in gtr. Next, see the first chapter of the book Gravitation and Inertia by Ciufolini and Wheeler, cited elsewhere on that webpage, for an intriguing but improperly formulated notion in terms of "distance weighted" contributions from distant matter to the local definition of a nonspinning frame. Then search the arXiv using the search bar at the main page of RelWWW for "Mach's principle" (this gives some twenty pages). Make sure to read the paper by Bondi enumerating dozens of forms of Mach principles! (Many of the other arXiv eprints discussing Mach principles are frankly not up to the standard set by Bondi, which is one reason why I suggest you start with this one.)

This is improperly formulated as stated. Try to avoid confusing "optical effects" with "clock effects" in relativistic physics.

"Time dilation" always refers to a comparison between two observers, so again this is improperly stated.

Last edited by a moderator: May 2, 2017
8. Apr 27, 2007

### MeJennifer

In other words, clocks on a rod go out of phase when this rod is accelerated. If instead you try to force all clocks to stay synchronized by applying the same acceleration for each slice of the rod orthogonal to the direction of acceleration, the space between these slices will increase and eventually the rod will break.

A webpage relevant for you might be from Mathpages http://www.mathpages.com/home/kmath422/kmath422.htm" [Broken]

Last edited by a moderator: May 2, 2017
9. Apr 28, 2007

### fireball3004

Okay, I think I understand what your saying Chris, your saying that acceleration (linear) is a curved path in space time right? But now I am confused on another subject, if it is acceleration because of it's curved path, does this mean it is not relative to other observers? I would have thought this would conflict with GR.

10. Apr 28, 2007

### Garth

In GR the geodesic between two events of a freely falling body is 'straight', in the sense that the interval is at an extremum but it is the space-time 'surface' on which the geodesic is 'drawn' that is 'curved'.

Thus inertial frames of reference accelerate relative to each other even though no forces are acting. In GR the gravitational force is re-interpreted as the effect of the curvature of space-time.

Garth

11. Apr 28, 2007

### Chris Hillman

Clarifying (?) what I did and did not say

No, I said that the acceleration vector, which is defined along the world line of some observer, is identified with the path curvature vector of that world line (treated as an arc length parameterized timelike curve), i.e. the covariant derivative of the unit tangent vector along the world line.

I did not say that acceleration is a curved path. What I did say does imply that timelike geodesics correspond to the world lines of observers in inertial motion.

The magnitude of acceleration equals the length of the covariant derivative of the unit tangent vector to the world line, so it is a scalar quantity defined along the world line and thus, not coordinate dependent.

You have probably been confused by reading popular accounts. It is however true that when Einstein set out on the path to general relativity, he had a number of desiderata, some of which are more completely realized in gtr than others.

12. Apr 29, 2007

### fireball3004

your saying that it is relative to the path through space time of the observer? And does this mean one can't see these effects of acceleration in the afore mentioned hypothetical?

13. Apr 29, 2007

### Chris Hillman

Huh?

It?

See?

Hypothetical?

14. Apr 29, 2007

### fireball3004

"it" = acceleration
"see" = observe
"hypothetical" = the situation I mentioned before where one observer is viewing an accelerating chain... note hypothetical can be used as a noun

15. Apr 30, 2007

### Demystifier

In general relativity, acceleration is absolute, and it is acceleration with respect to metric field. Einstein equation has nonzero solutions for the metric even in the absence of matter. The solution is not unique, but depends on the choice of initial conditions. In this way, a "preferred" class of inertial frames is determined by the choice of the initial condition.