Abstract
After studying the general formalism of tensors, we can now apply it specifically to the 4-dimensional spacetime arena. A fundamental addition to the general baggage of tensors is the causal character of 4-vectors in Minkowski spacetime, which is due to the Lorentzian signature of the Minkowski metric. As a consequence of this signature, the line element is not positive-definite and we already know that this feature is linked to the existence of light cones, which play a crucial role because they determine the causal structure of the theory. Moreover, some specific notation which was not used in the general discussion of tensors applies to the Minkowski spacetime of Special Relativity.
Make everything as simple as possible, but no simpler.
—Albert Einstein.
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Notes
- 1.
The symbol \(\dot{=}\) denotes equality in a particular coordinate system.
- 2.
This is the reason why, so far, we have not distinguished between contravariant components \(A^i\) and covariant components \(A_i\) of a vector \(\mathbf{A}\) in \(\mathbb R ^3\) with Cartesian coordinates.
- 3.
The definition of the d’Alembertian \(\nabla ^{\mu }\nabla _{\mu }\phi \) in general coordinates requires the notion of covariant derivative \(\nabla _{\alpha }\) introduced in Chap. 10.
- 4.
It is an old Popperian adage that a theory cannot be verified: it can only be falsified.
References
L.D. Landau, E. Lifschitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1989)
R.M. Wald, General Relativity (Chicago University Press, Chicago, 1984)
S.M. Carroll, Spacetime and Geometry, An Introduction to General Relativity (Addison-Wesley, San Francisco, 2004)
R. d’Inverno, Introducing Einstein’s Relativity (Clarendon Press, Oxford, 2002)
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Faraoni, V. (2013). Tensors in Minkowski Spacetime. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01107-3_5
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DOI: https://doi.org/10.1007/978-3-319-01107-3_5
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