Does a relativistic rolling ball wobble?

1effect

You are misquoting me, see the correct quote here.

"There are two effects at work , one is the well-known relativistic length contraction, the other one is the difference in arrival of light rays. "

As an aside, "light taking longer to reach the observer from the parts of the ball furthest from the observer" and "the Terrell effect" are one and the same thing :-)

Yes, this is true, I pointed that out in one of my posts. I also mentioned that it is not clear why this is relevant.

Are you saying that the axis of rotation describes a cycloid? I would be very interested in the proof.

JesseM

How is that a difficulty? If you know the movement of the point in the frame of the center, the Lorentz transformation will tell you the time of any event on that point's worldline in another frame. It's really just a problem of the equations being difficult or impossible to solve exactly. For example, if the point on the rim is described by x'(t') = cos(t') and y'(t') = sin(t') in the rest frame of the center, applying the Lorentz transformation would give the equations:

gamma*(x - vt) = cos(gamma*(t - vx/c^2))
y = sin(gamma*(t - vx/c^2)

The difficulty is just in solving the first equation for x (if you could do that, you could plug the answer into the second equation to get y(t)). There may not be an exact solution, I don't know. But if you're just interested in a visual depiction of the path, it's an easy enough matter to just pick a bunch of different t' coordinates, find the corresponding x' and y' coordinates for each one, then convert each (x', y', t') to an (x, y, t) using the Lorentz transform.

1effect

Actually, it is possible to get a parametric solution, avoiding the nasty transcendental equations. Here is the trick:

[tex]x'^2+y'^2=1[/tex]

[tex]\gamma^2 (x-vt)^2+y^2=1[/tex]

[tex]y=sin(t)[/tex]
[tex]\gamma (x-vt)=cos(t)[/tex]
so:
[tex]x=\gamma^{-1} cos(t)+vt[/tex]

JesseM

How did you get y=sin(t)? Wasn't the equation in the primed frame y'=sin(t'), which means it should be y=sin(gamma*(t - vx/c^2))

1effect

I parametrized :

[tex]\gamma^2 (x-vt)^2+y^2=1[/tex]

There is an indefinite number of such parametrizations, they all produce the same curve.

For example, I could have chosen [tex]\frac{1-t^2}{1+t^2}[/tex] and [tex]\frac{2t}{1+t^2}[/tex]

JesseM

I don't understand what you mean by "re-parametrized" here. Can you fill in the missing steps that allow you to go from that equation to y=sin(t)? Unless you mean that y=sin(t) is meant to be the definition of a new parameter t, but then you should have given the parameter a different name than the time coordinate that appears in (x-vt).

JesseM

But if you're just parametrizing y with a newly-defined parameter, you can't use the t-coordinate as the parameter! The t-coordinate already has a specific meaning--if you have some y' and t' which lie on the worldline of the point in the primed frame, you can't substitute in y=sin(t) unless it's true that when you do the Lorentz transform on that y' and t' on the point's worldline, the resulting y and t actually satisfy y=sin(t). If you want to define a new parameter you should give it a different name, like y=sin(p).

1effect

I am not sure about that , I will need to think about it.
This would result into :

[tex]y=sin(\tau)[/tex]

[tex]x=\gamma^{-1} cos(\tau)+vt[/tex]

JesseM

But that isn't very helpful, because you don't know what value of t corresponds to a given value of [tex]\tau[/tex].

1effect

Yes, this wouldn't work. Too bad, I guess we are stuck with the unsolvable system of transcendental equations. :-(

Antenna Guy

When you guys are done "simplifying" the problem, I'd like to see what you get when your observer is not at infinity.

Consider that the sphere travelling along x at relativistic speed with respect to an inertial observer will be described by:

[tex]\frac{\gamma^2 x'^2}{r^2}+\frac{y'^2}{r^2}+\frac{z'^2}{r^2}=1[/tex]

[note: sorry about leaving out the velocity term - I'd have probably screwed that up anyway ]

Where r is the radius of the sphere in its' rest frame.

If the inertial observer is not at infinity, the "shape" of the object will not be a projection on a plane - and that "shape" will vary with distance/aspect angle.

If you get that far, you can let the surface rotate about its' volume...

Regards,

Bill

JesseM

At infinity? I wasn't trying to calculate what would be seen visually, just the coordinate path of a point on the rim in a frame where the rolling object was moving.
I guess this would be the equation for the coordinates of the surface in the frame where the sphere is moving along the x' axis, at the moment the sphere was centered at the origin. But this is different from the shape the sphere would appear to be as seen by an observer at rest in this frame.
What do you mean by "projection on a plane"?

Phrak

Did I miss it, or hasn't it been pointed out? It's not at all unambiguous as to what meant by a 'ball'?

First, is it a problem in dynamics, or simply kinematics? If only kinematics, the spinning ball has a circumference to diameter ratio of gamma*pi in the plane of angular momentum. So in this case, it's fair to ask 'was it a ball before spinning? (If it was a ball before spinning then its a ball of constant radius, otherwise constant circumference.)

Antenna Guy

Are you aware that this relativistic 'ball' does not necessarily have an instantaneous circumference (per se) in the frame of the observer?

Let's say that the observed circumference of this ball spans a range of time that approaches [itex]\frac{2r}{c}.[/itex]. What would you integrate over to verify that the rest length of the observed circumference is still [itex]2\pi r[/itex]?

Regards,

Bill

1effect

The above is not the equation of the moving sphere. You need to apply to Lorentz transform correctly.

JesseM

I think it is the equation for the sphere's surface at a given instant when it is centered at the origin and moving on the x'-axis. The equation describes an ellipsoid with radius r along the y' and z' axis, but Lorentz-contracted radius [tex]r/\gamma[/tex] along the x'-axis...is this not correct?

1effect

"moving" was key in my answer.
His equation is the snapshot for t=0, so we agree.

The correct equation is:

[tex]\frac{\gamma^2 (x'-vt')^2}{r^2}+\frac{y'^2}{r^2}+\frac{z'^2}{r^2}=1[/tex]