Galilean Vs. Lorentz Transformations

[rmiller70015]

I was reading my textbook for my elementary modern class and the author said that a pulse of light from a light bulb would be spherical and could be expressed as x² + y² + z² = c²t² and x'² + y'² + z'² = c²t'². Then the author goes on to say that this cannot happen for both reference frames in a Galilean Transform. Why can't both frames have a spherical wave front?

 

[Ibix]

They can. They can't both have a spherical wavefront centered on the origin, which is what the source and frame independence of the speed of light requires, and what your two expressions say.

 

[PeroK]

I was reading my textbook for my elementary modern class and the author said that a pulse of light from a light bulb would be spherical and could be expressed as x² + y² + z² = c²t² and x'² + y'² + z'² = c²t'². Then the author goes on to say that this cannot happen for both reference frames in a Galilean Transform. Why can't both frames have a spherical wave front?

If space and time are absolute, then you could draw both views of the expanding light sphere on the same diagram - just using two spatial dimensions, say. As one observer is moving wrt to other, at most one can remain at the centre of a circle/sphere. A sphere cannot have two different centres. Not without the relativity of lengths and times.

 

[pixel]

The two equations you show are for the case where there are two frames of reference and a light pulse is emitted at their origins when the origins coincide. Say that the light bulb is at rest in the unprimed frame and the primed frame is moving to the right at some speed v < c. In a Galilean transformation, the speed of the light ahead of the primed origin would be c - v, since the origin is catching up with the light to some extent, and behind the primed origin it would be c + v, since the origin is moving away from the light. As it's not the same in both directions in the primed frame, it won't be part of a spherical wave front.

 

[Ibix]

The two equations you show are for the case where there are two frames of reference and a light pulse is emitted at their origins when the origins coincide. Say that the light bulb is at rest in the unprimed frame and the primed frame is moving to the right at some speed v < c. In a Galilean transformation, the speed of the light ahead of the primed origin would be c - v, since the origin is catching up with the light to some extent, and behind the primed origin it would be c + v, since the origin is moving away from the light. As it's not the same in both directions in the primed frame, it won't be part of a spherical wave front.

This is not correct. The surface will be spherical in both frames. Galilean transforms don't alter the shape of things. The surface ##x^2+y^2+z^2=c^2t^2## transforms to ##(x'+vt')^2+y'^2+z'^2=c^2t'^2##, which is simply a sphere centered on ##x'=-vt'## - i.e. a moving sphere.

 

[Bartolomeo]

In classical case measured by observer frequencies will be different, depending whether he is in motion towards center of the sphere, or sphere is in motion towards him.

In first case (classical) if he moves towards centre of the sphere with a speed of light, maximum frequency he observes will be 2f.

In second case (classical, sphere moves towards him) all wave fronts will hit him instantly and measured frequency will be infinitely high.

However, time dilation (Lorentz factor) makes measured frequencies equivalent in relativistic case. In first case we assign Lorentz factor to the observer and his clock dilates, so measured frequency tends to infinitely large value as he approaches that of light. He perceives that wave fronts cross him rarely than in classical case since his own clock dilates.

In second case we assign Lorentz factor to the source. Source’s clock dilates and source of waves oscillates rarely. That leads to redshift (i.e. measured blueshift will be less blue, than it should be in classical case), which makes observations reciprocal.

The same reflections will be in case, if observer and centre of the sphere fly away from each other.

Thus, it is exactly Lorentz factor (or requirement of equivalence of speed of light for all observers) what makes sphere to be a sphere in all reference frames. It seems like that.

Spherical wavefronts of light

https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations

We can also assign Lorentz factor to each of them in equal proportions. But I am not sure, that we can assign full amount of Lorentz factor to both of them.

 

[Ibix]

In classical case measured by observer frequencies will be different, depending whether he is in motion towards center of the sphere, or sphere is in motion towards him.

In first case (classical) if he moves towards centre of the sphere with a speed of light, maximum frequency he observes will be 2f.

In second case (classical, sphere moves towards him) all wave fronts will hit him instantly and measured frequency will be infinitely high.

You are assuming, here, that "classical light" travels in a medium at c with respect to that medium. You are then talking about two different experiments, one where the source is at rest with respect to the medium and one where it is in motion with respect to the medium. This is not the same as the passive frame changes of the Galilean or Lorentz transforms, which simply switch between different descriptions of the same experiment.

 

[Bartolomeo]

You are assuming, here, that "classical light" travels in a medium at c with respect to that medium. You are then talking about two different experiments, one where the source is at rest with respect to the medium and one where it is in motion with respect to the medium. This is not the same as the passive frame changes of the Galilean or Lorentz transforms, which simply switch between different descriptions of the same experiment.

Passive frame change in classical case introduces two different spheres. In relativistic case we consider the same sphere. Sure we can passive change frames and to get the same outcome.
Sphere and experiment is the same, but explanations are different and frame dependent.

 

[PeroK]

Passive frame change in classical case introduces two different spheres. In relativistic case we consider the same sphere. Sure we can passive change frames and to get the same outcome.

This is not correct. In the classical case, there is and can only be one sphere, centred on one origin. The Galilean transformation must have the second origin ##O'## not at the centre of the sphere.

With a Lorentz Transformation, an expanding light sphere from origin ##O, t## can also be an expanding light sphere from origin ##O', t'##, even though the orgins only instantaneously coincide at ##t= t' =0##.

 

[pixel]

This is not correct. The surface will be spherical in both frames. Galilean transforms don't alter the shape of things. The surface ##x^2+y^2+z^2=c^2t^2## transforms to ##(x'+vt')^2+y'^2+z'^2=c^2t'^2##, which is simply a sphere centered on ##x'=-vt'## - i.e. a moving sphere.

Whoops. Thanks for the correction. Not much can get by here on PF.