Relativistic Sphere: Length Contraction & Volume

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• Einstein's Cat
In summary, assuming a sphere is at rest in a particular frame and then velocity changes, the sphere's radius will contract in the direction of one dimensional motion.

Einstein's Cat

Say there is a theoretical sphere of radius r, at rest, then if it's velocity changes then I assume that the radius is subject to length contraction and thus it's volume would decrease from a stationary observer. Is this assumption true?

http://www.spacetimetravel.org/fussball/fussball.html

The sphere becomes an ellipsoid but due to the finite speed of light and the fact that the light travels different distances to reach your eye you will still see a sphere.

A similar case occurs with a relativistic cube where the cube contracts relative to the stationary observer but the observer actually sees a cube rotated slight toward him/her so that you see the side and the back together because light from the fornt edge gets to your eyes sooner than light from the back edge.

http://www.spacetimetravel.org/tompkins/node1.html

Look up Terrell rotation for more details:

http://www.math.ubc.ca/~cass/courses/m309-01a/cook/terrell1.html

Einstein's Cat
Einstein's Cat said:
Say there is a theoretical sphere of radius r, at rest, then if it's velocity changes then I assume that the radius is subject to length contraction and thus it's volume would decrease from a stationary observer. Is this assumption true?

It's easier if you consider a sphere at rest in it's own reference frame, and asks what happens if you ask what it's shape is in some frame moving relative to the sphere's rest frame. That way you don't have to worry about the notion of "rigidity".

The sphere when seen from a moving frame does indeed Lorentz contract in one direction, assuming the shape of an ellipsoid. The volume of the sphere in the moving reference frame is lower, volume is therefore a frame-dependent quantity like length is.

pervect said:
It's easier if you consider a sphere at rest in it's own reference frame, and asks what happens if you ask what it's shape is in some frame moving relative to the sphere's rest frame. That way you don't have to worry about the notion of "rigidity".

The sphere when seen from a moving frame does indeed Lorentz contract in one direction, assuming the shape of an ellipsoid. The volume of the sphere in the moving reference frame is lower, volume is therefore a frame-dependent quantity like length is.
What is the relationship between the volume of the ellipsoid and its radius?

The ones in the y and z direction don't change. The one in the x direction scales as ##\gamma##.

Ibix said:
the x direction scales as ##\gamma##.
I see because length contraction is in the direction of one dimensional motion. I apologise but what does the quote above mean?

What you said. Although I should have said ##1/\gamma##. At speed v the length is ##\sqrt {1-v^2/c^2}## times its rest length.

Einstein's Cat
Ibix said:
x direction scales as ##\gamma##.
Sorry to go on but when you say scale do you perhaps mean that "x new"
= 1/Lamda * "x rest"?

Einstein's Cat said:
Sorry to go on but when you say scale do you perhaps mean that "x new"
= 1/Lamda * "x rest"?
That's a gamma (##\gamma##), not a lambda (##\lambda##), but otherwise yes.

Ibix said:
That's a gamma (##\gamma##), not a lambda (##\lambda##), but otherwise yes.
Thank you very much and I'll go and revise greek letters!

1. What is a relativistic sphere?

A relativistic sphere is a theoretical concept in physics that describes a spherical object moving at a significant fraction of the speed of light. This concept is used to explain the effects of length contraction and volume compression that occur at high speeds according to Einstein's theory of relativity.

2. What is length contraction in a relativistic sphere?

Length contraction is the phenomenon where an object appears shorter in the direction of its motion when moving at high speeds. In a relativistic sphere, this effect is observed as the radius of the sphere appears shorter to an observer moving at a different speed than the sphere.

3. How does volume change in a relativistic sphere?

In a relativistic sphere, the volume of the sphere appears to decrease due to the effects of length contraction. This means that the sphere will appear smaller in all dimensions to an observer moving at a different speed than the sphere.

4. What is the formula for calculating length contraction in a relativistic sphere?

The formula for calculating length contraction in a relativistic sphere is L = L0 * √(1 - v2/c2), where L is the contracted length, L0 is the rest length, v is the relative velocity, and c is the speed of light.

5. Can the effects of length contraction and volume compression be observed in everyday life?

No, the effects of length contraction and volume compression are only significant at speeds close to the speed of light. In everyday life, objects are not moving at such high speeds, so these effects are not noticeable. They are only observed in experiments involving high-speed particles or in astronomical observations.

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