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Fundamentals of Special Relativity

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Special Relativity

Part of the book series: Undergraduate Lecture Notes in Physics ((ULNP))

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Abstract

The theory of Special Relativity formulated by Albert Einstein in 1905 [1, 2] is, without doubt, one of the great intellectual achievements of the 20th century. Our everyday experience is about objects moving at speeds much smaller than the speed of light in vacuo, \({ c =2.99792458 \times 10^{8} \; \mathrm {m}/\mathrm {s}}\). Newtonian mechanics was developed to describe phenomena at typical speeds \({ v \ll c}\) and fails when speeds are not negligible in comparison with \(c\). This situation is not infrequent; for example, it is relatively easy to accelerate electrons to speeds \({ v = 0.99 \,c}\) in accelerators. However, as their velocities become closer and closer to \(c\), it becomes harder and harder to accelerate these electrons further.

Concepts that have proven useful in ordering things easily achieve such authority over us that we forget their earthly origins and accept them as unalterable givens.

—Albert Einstein.

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Notes

  1. 1.

    See Ref. [3] for a brief outline of the ideas leading to the development of Special and General Relativity and, e.g., Ref. [4] for a discussion of the limitations of the ideas of Henri Poincaré and Hendrik-Antoon Lorentz in the development of Special Relativity.

  2. 2.

    Exceptions are certain theories of quantum gravity, but these are completely speculative and probing them does not seem feasible in the near future.

  3. 3.

    Remember that, in Cartesian coordinates, the electric and magnetic fields have components \({ \mathbf E =\left( E^{x},E^{y},E^{z}\right) }\) and \( { \mathbf B =\left( B^x ,B^y,B^z\right) } \), \({ \mathbf \nabla \cdot \mathbf E \equiv \frac{\partial E^{x}}{\partial x} + \frac{\partial E^{y}}{\partial y} + \frac{\partial E^{z}}{\partial z} }\), \( { \mathbf \nabla \times \mathbf B \equiv \left| \begin{array}{ccc} \hat{i} &{} \hat{j} &{} \hat{k}\\ \partial _{x} &{} \partial _{y} &{} \partial _{z}\\ B^{x} &{} B^{y} &{} B^{z} \end{array} \right| } \), and \( { \frac{\partial \mathbf E }{\partial t}\equiv \left( \frac{\partial E^{x}}{\partial t}, \frac{\partial E^{y}}{\partial t},\frac{\partial E^{z}}{\partial t}\right) } \).

  4. 4.

    Today the concept of ether seems artificial even at first sight, but it was not so unnatural in the mechanistic views of the physicists of the 1800s.

  5. 5.

    The ray traveling in the vertical direction is oblique because of reflection off the beam splitter which is moving with respect to the ether (see Refs. [7–11] for details).

  6. 6.

    It appears that Michelson and Morley’s original paper contained an error of order higher than \((v/c)^2\) [12]. The detailed higher order analysis of the Michelson–Morley experiment is rather complicated [13, 14], but it is not important here.

  7. 7.

    Nowadays, the experiment is regarded as a test of the isotropy of the speed of light [15], which has been probed with laser versions of the Michelson–Morley experiment with an accuracy of \(10^{-15}\) [16].

  8. 8.

    However, the idea of motion-dependent forces is not completely incorrect: one can make a toy model of a solid with point charges bound by electrostatic forces. The latter transform in a known way under a change of inertial frame giving effectively a contraction in the direction of motion, the same which is obtained by length contraction of the inter-particle separations [19].

  9. 9.

    However, only the gravitational and electromagnetic interactions were known at the beginning of the twentieth century.

  10. 10.

    It appears that, in the early writings on Special Relativity, the constancy of the speed of light was mostly intended as meaning that the speed of light is independent of the velocity of the source, not of the inertial frame in which it is measured [20].

  11. 11.

    Indeed, interplanetary distances are measured using electromagnetic waves reflected off a target.

  12. 12.

    There is a long tradition of deriving special-relativistic formulas from gedankenexperimenten [21].

  13. 13.

    The argument holds because of the finiteness of the speed of light, in contrast with Newtonian mechanics in which the latter is infinite. It is also essential that the speed of light is the same in the two inertial frames.

  14. 14.

    The twin “paradox” has been discussed since the early days of Special Relativity by many people, including Paul Langevin, Albert Einstein, and Max Born. The Hafele-Keating experiment [25] can be regarded as a direct experimental verification of the twin “paradox”.

  15. 15.

    Locally, acceleration is equivalent to a gravitational field and is properly treated in General Relativity.

  16. 16.

    Interestingly, if the \({\left( x, ct \right) } \) world is curled into a cylinder so that the “rocket twin” John does not have to invert his course to come back to earth because he describes a closed curve in this closed universe, the asymmetry persists and the ages of the twins still differ [26].

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  1. Physics Department, Bishop’s University, College Street 2600, Sherbrooke, QC, J1M 1Z7, Canada

    Valerio Faraoni

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Faraoni, V. (2013). Fundamentals of Special Relativity. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01107-3_1

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