qupta

03-25-2010, 10:58 AM

Twin Rigid Body Spheres Contradiction

Assume we have two rigid body spheres of rest radius r in relative motion v along the x-axis. O is the stationary sphere and O' is the moving sphere. The moving sphere has a light source and clock at its origin.

The stationary sphere has a clock at its origin.

Assume when the origins of the two spheres are co-located, the two clocks are synched, and the light is flashed.

The following are the deductions from SR.

Light must proceed spherically from the center of the O sphere and strikes the rigid body sphere simultaneously.

Light must proceed spherically from the center of the O' sphere and strikes the rigid body sphere simultaneously.

O' does not believe O is struck simultaneously.

O does not believe O' is struck simultaneously.

For an observer sitting at O, when r/c elapses on its clock, all rigid body sphere points are struck simultaneously.

For an observer sitting at O', when r/c elapses on its clock, all rigid body sphere points are struck simultaneously.

Now, allow the clock at O to elapse a time of rγ/c after the flash of light. This implies light has proceeded a distance rγ in all directions from O. In addition, by time dilation, the clock at O' elapsed r/c. However, when the clock of O' elapses r/c, all of its sphere points are struck simultaneously. This implies light is a distance r in all directions from the center O'.

Now, only the positive x-axis light beam will be considered.

According to LT, x' = (x- vt)γ.

Since t = rγ/c and x = rγ, then x' = (rγ- v(rγ/c))γ = r(γ² - vγ²/c).But, since the clock at O elapsed rγ/c, then the clock at O' elapsed r/c and hence, x' = r since O' must meet is rigid body sphere simultaneity requirements when its clock reads r/c.

Therefore, under the rules of SR, when the clock of O elapses rγ/c, the one light beam is located at both x' = r and x' = r(γ² - vγ²/c) which is a contradiction.

Alternatively, when the clock at O elapses rγ/c, light is a distance rγ in all directions as stated above. In addition, the clock of O' must elapse r/c because of time dilation. Since the time elapsed r/c on the clock of O', then light must be a distance r from O' in all directions and in particular along the positive x-axis.By LT, x = (x' + vt')γ.

Since x' = r and t' = r/c, then x = (r + vr/c)γ =* rγ(1 + v/c).Hence, along the positive x-axis, light is a distance rγ from O, while at the same time, light is a distance rγ(1 + v/c) from O which is a contradiction.**

Assume we have two rigid body spheres of rest radius r in relative motion v along the x-axis. O is the stationary sphere and O' is the moving sphere. The moving sphere has a light source and clock at its origin.

The stationary sphere has a clock at its origin.

Assume when the origins of the two spheres are co-located, the two clocks are synched, and the light is flashed.

The following are the deductions from SR.

Light must proceed spherically from the center of the O sphere and strikes the rigid body sphere simultaneously.

Light must proceed spherically from the center of the O' sphere and strikes the rigid body sphere simultaneously.

O' does not believe O is struck simultaneously.

O does not believe O' is struck simultaneously.

For an observer sitting at O, when r/c elapses on its clock, all rigid body sphere points are struck simultaneously.

For an observer sitting at O', when r/c elapses on its clock, all rigid body sphere points are struck simultaneously.

Now, allow the clock at O to elapse a time of rγ/c after the flash of light. This implies light has proceeded a distance rγ in all directions from O. In addition, by time dilation, the clock at O' elapsed r/c. However, when the clock of O' elapses r/c, all of its sphere points are struck simultaneously. This implies light is a distance r in all directions from the center O'.

Now, only the positive x-axis light beam will be considered.

According to LT, x' = (x- vt)γ.

Since t = rγ/c and x = rγ, then x' = (rγ- v(rγ/c))γ = r(γ² - vγ²/c).But, since the clock at O elapsed rγ/c, then the clock at O' elapsed r/c and hence, x' = r since O' must meet is rigid body sphere simultaneity requirements when its clock reads r/c.

Therefore, under the rules of SR, when the clock of O elapses rγ/c, the one light beam is located at both x' = r and x' = r(γ² - vγ²/c) which is a contradiction.

Alternatively, when the clock at O elapses rγ/c, light is a distance rγ in all directions as stated above. In addition, the clock of O' must elapse r/c because of time dilation. Since the time elapsed r/c on the clock of O', then light must be a distance r from O' in all directions and in particular along the positive x-axis.By LT, x = (x' + vt')γ.

Since x' = r and t' = r/c, then x = (r + vr/c)γ =* rγ(1 + v/c).Hence, along the positive x-axis, light is a distance rγ from O, while at the same time, light is a distance rγ(1 + v/c) from O which is a contradiction.**