Summary Lorentz Transformations No Invariant Speed Required arxiv.org
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One Line
The derivation of Lorentz transformations does not assume an invariant speed, but rather relies on electromagnetism to determine the velocity scale as the speed of light.
Slides
Slide Presentation (7 slides)
Key Points
- The Lorentz transformations can be derived without assuming an invariant speed
- The Lorentz transformations follow from the absence of privileged inertial reference frames and the group structure of the transformations
- The existence of an invariant speed is a consequence of the principle of relativity, not a necessary assumption
- Von Ignatowsky derived the result that the Lorentz transformations are the only admissible transformations consistent with certain principles
- The Lorentz transformations are not directly related to the properties of electromagnetic radiation
Summaries
19 word summary
Lorentz transformations are derived without assuming an invariant speed. Electromagnetism determines the velocity scale as the speed of light.
71 word summary
The Lorentz transformations can be derived without assuming an invariant speed. Von Ignatowsky showed that the only transformations consistent with the principle of relativity are the Lorentz transformations or the Galilei transformations. Electromagnetism fixes the velocity scale, identified as the speed of light. The authors present an elementary proof of von Ignatowsky's result. The Lorentz transformations and Galilei transformations are the only structures compatible with the principles of inertia and relativity.
181 word summary
The Lorentz transformations can be derived without assuming an invariant speed. They are a consequence of the principle of relativity and not directly related to the properties of electromagnetic radiation. Von Ignatowsky showed that the only transformations consistent with the principle of inertia, isotropy of space, absence of preferred inertial frames, and closure under composition are the Lorentz transformations or the Galilei transformations. Electromagnetism is relevant in fixing the arbitrary velocity scale, which is identified as the speed of light. The authors present an elementary proof of von Ignatowsky's result suitable for an introductory course in special relativity. The transformations between inertial frames are characterized using space and time coordinates, and the coefficients of the transformation are constrained by requirements related to the principle of inertia and relativity. The authors also derive conditions for the matrix representing the transformation, finding three possibilities: Galilean transformations, Lorentz transformations with an arbitrary velocity scale, and Lorentz transformations with the speed of light as the velocity scale. The Lorentz transformations and Galilei transformations are the only structures compatible with the principles of inertia and relativity.
402 word summary
The Lorentz transformations can be derived without assuming the existence of an invariant speed. This is based on the absence of privileged inertial reference frames and the group structure of the transformations. The Lorentz transformations are a consequence of the principle of relativity, not directly related to the properties of electromagnetic radiation. However, the finite value of the speed of light must be obtained from experiment.
Von Ignatowsky's approach to deriving the Lorentz transformations is different. He showed that the only admissible transformations consistent with the principle of inertia, isotropy of space, absence of preferred inertial frames, and closure under composition are the Lorentz transformations or the Galilei transformations. Electromagnetism is only relevant in fixing the arbitrary velocity scale, which is then identified with the speed of light.
The authors present an elementary proof of von Ignatowsky's result suitable for an introductory course in special relativity. They characterize the transformations between two inertial frames using space and time coordinates, simplifying the argument by considering only x and t coordinates. The principle of inertia, stating that free particles undergo rectilinear motion with constant speed in an inertial frame, implies that the two frames are related by a linear transformation.
The authors express the hypothesis of no privileged frames by considering two events at the same spatial location in one frame and separated by a time difference. They require that if two different events at the same spatial location in another frame are separated by the same time difference, then these events are separated by the same time lapse in the first frame. They also require that if a rod is at rest in one frame and has a certain length, then the same result is obtained by another frame for an identical rod at rest. These requirements constrain the coefficients of the transformation.
To further constrain the structure of the transformation, the authors add the requirement that the transformations connecting two inertial frames form a group. They derive conditions for the matrix representing the transformation by considering the combination of two transformations. Three possibilities are found: Galilean transformations, Lorentz transformations with an arbitrary velocity scale, and Lorentz transformations with the speed of light as the velocity scale.
In conclusion, the Lorentz transformations and Galilei transformations are the only structures compatible with the principles of inertia and relativity. The authors provide an elementary derivation of this result applicable to undergraduate courses on special relativity.
470 word summary
The Lorentz transformations can be derived without assuming the existence of an invariant speed. The structure of the transformations is based on the absence of privileged inertial reference frames and the group structure of the transformations. This result was first derived by von Ignatowsky in 1911. The Lorentz transformations are not directly related to the properties of electromagnetic radiation, but rather are a consequence of the principle of relativity. However, the finite value of the speed of light must be obtained from experiment.
The Lorentz transformations are usually derived from the invariance of the speed of light, which implies the invariance of the Minkowski interval. Von Ignatowsky's approach is different and does not assume the existence of an invariant speed. He showed that the only admissible transformations consistent with the principle of inertia, isotropy of space, absence of preferred inertial frames, and closure under composition are the Lorentz transformations or the Galilei transformations. Electromagnetism is only relevant in fixing the arbitrary velocity scale, which is then identified with the speed of light.
The authors present a completely elementary proof of von Ignatowsky's result that is suitable for an introductory course in special relativity. They characterize the transformations between two different inertial frames using space and time coordinates. They simplify the argument by considering transformations involving only x and t coordinates. They assume the validity of the principle of inertia, which states that free particles undergo rectilinear motion with constant speed in an inertial frame. This implies that the two inertial frames are related by a linear transformation.
The authors express the hypothesis of no privileged frames in a more transparent way by considering two events at the same spatial location in one frame and separated by a time difference. They require that if two different events at the same spatial location in another frame are separated by the same time difference, then these events are separated by the same time lapse in the first frame. They also require that if a rod is at rest in one frame and has a certain length, then the same result is obtained by another frame for an identical rod at rest. These requirements constrain the coefficients of the transformation.
To further constrain the structure of the transformation, the authors add the requirement that the transformations connecting two inertial frames form a group. They consider the combination of two transformations and derive conditions for the matrix representing the transformation. They find three possibilities: Galilean transformations, Lorentz transformations with an arbitrary velocity scale, and Lorentz transformations with the speed of light identified as the velocity scale.
In conclusion, the Lorentz transformations and Galilei transformations are the only structures compatible with the principles of inertia and relativity. The authors provide an elementary derivation of this result that can be used in undergraduate courses on special relativity.