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The General Theory of Relativity pp 105–228Cite as

The Pseudo-Riemannian Space-Time Manifold M4

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Abstract

We shall provide here a brief review of the special theory of relativity. Since the emphasis of this book is general relativity, we will state some of the results in this section without proof, for future application. For more details, the reader is referred to the references provided within this chapter.

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Notes

  1. 1.

    There are strong physical arguments for this assumption which are beyond the scope of this brief review of special relativity. The interested reader is referred to [55].

  2. 2.

    Often, the term “constant acceleration” in the relativity literature refers to the condition \({d}_{ij}\frac{\mathrm{d}{\mathcal{U}}^{i}(s)} {\mathrm{d}s} \frac{\mathrm{d}{\mathcal{U}}^{j}(s)} {\mathrm{d}s} = \mbox{ const.}\), which may be satisfied with nonconstant \(\frac{\mathrm{d}{\mathcal{U}}^{i}(s)} {\mathrm{d}s} = \frac{{\mathrm{d}}^{2}{{\mathcal {X}}}^{{\#}i}(s)} {\mathrm{d}{s}^{2}}\).

  3. 3.

    We are using Lorentz–Heaviside units. Especially, we have set c, the speed of light, to be unity to be consistent with units used throughout most of the book.

  4. 4.

    The equivalency between gravitational mass and inertial mass is assumed. Current experimental limits place the equivalency of these two types of mass to within 10 − 12 [226], and it is widely believed that they are equivalent.

  5. 5.

    The apparent zero-gravity effects experienced by astronauts in orbit are exactly analogous to the elevator example. The “zero-gravity” effects are due to the fact that the astronauts and the orbiting vehicle are in free fall and not due to the fact that gravity is too weak to have any appreciable effects. A quick calculation, using (2.79ii) and the remark after Example 1.3.28, reveals that the gravitational acceleration at 320 km above the Earth’s surface (low Earth orbit) is approximately 91% of its value at the surface of the Earth.

  6. 6.

    Consult Appendix 3 for the definition of a Segre characteristic.

  7. 7.

    Let \(\vec{\mathbf{V}}(\cdot \cdot )\) be a continuous vector field defined on Σ such that \(\vec{\mathbf{V}}(\cdot \cdot )\neq \vec{\mathbf{0}}(\cdot \cdot )\) for any point x on ∂Σ. If the index or winding number of \(\vec{\mathbf{V}}(\cdot \cdot )\) around ∂Σ is not zero, then there exists at least one x 0 ∈ Σ such that \(\vec{\mathbf{V}}({x}_{0}) =\vec{ \mathbf{0}}({x}_{0})\,\). (See [37].)

  8. 8.

    . The Helmotz theorem [159] on differentiable vector field \(\vec{\mathbf{W}}(\mathbf{x})\) in a three-dimensional domain

    allows the decomposition

    $${W}^{\alpha }(\mathbf{x}) = {\nabla }^{\alpha }h + {\eta }^{\alpha \beta \gamma }(\mathbf{x})\left [{\nabla }_{\!\gamma }{A}_{ \beta } -{\nabla }_{\!\beta }{A}_{\alpha }\right ].$$
    1. 2.

      For a closed, differential p-form of (1.58), the Hodge decomposition theorem [104] asserts that

      $$\begin{array}{rcl}{ W}_{{i}_{1},\ldots,{i}_{p}}(x)\,\mathrm{d}{x}^{{i}_{1}} \wedge \ldots \wedge \mathrm{d}{x}^{{i}_{p}} =\;& {h}_{{ i}_{1},\ldots,{i}_{p}}(x)\,\mathrm{d}{x}^{{i}_{1}} \wedge \ldots \wedge \mathrm{d}{x}^{{i}_{p}} & \\ \;& +\mathrm{d}\left [{\alpha }_{{i}_{1},\ldots,{i}_{p-1}}(x)\,\mathrm{d}{x}^{{i}_{1}} \wedge \ldots \wedge \mathrm{d}{x}^{{i}_{p-1}}\right ],& \\ {\nabla }_{\!j}{\nabla }^{j}{h}_{{ i}_{1},\ldots,{i}_{p}} =\;& 0. & \\ \end{array}$$
  9. 9.

    In an N-dimensional domain, at most N (reasonable), coordinate conditions hold. Therefore, a two-dimensional metric can be locally reduced to a conformally flat form. A three-dimensional metric can be locally brought to an orthogonal form. However, orthogonal coordinates may not exist in dimensions N > 3. The coordinate conditions \(\widehat{{\mathcal{C}}}^{m}\left (\widehat{{g}}_{ij}\left (\widehat{x}\right )\right ) = 0\) are not tensor field equations.

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    Anadijiban Das & Andrew DeBenedictis

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Das, A., DeBenedictis, A. (2012). The Pseudo-Riemannian Space-Time Manifold M4. In: The General Theory of Relativity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3658-4_2

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