Abstract
The principle of relativity requires that the laws of physics have the same mathematical expression in all inertial coordinate systems. When applied to special relativity, this “law of laws” requires that the mathematical expressions for the laws of physics be Lorentz covariant. Therefore, when formulating physics in the framework of special relativity, all the known laws of physics should be inspected; those that satisfy this requirement remain laws, while those that do not must be reformed until they meet this criterion. First, we inspect Maxwell’s theory of electromagnetism. Maxwell’s equations are endowed with Lorentz covariance (which can be seen more explicitly in its 4-dimensional formulation, see Sect. 6.6), and thus can be integrated into the framework of special relativity without being reformed. This is in fact not strange at all, since one of the important reasons special relativity came about is that Maxwell’s theory contradicts the notion of pre-relativity spacetime. Next, we will inspect Newton’s laws of motion. As an example, consider the law of conservation of momentum. As we pointed out at the beginning of Sect. 6.3, if the definition of momentum \(\vec {p}=m\vec {u}\) is still used, then conservation of momentum violates Lorentz covariance and must be modified. By redefining momentum as \(\vec {p}=m\vec {u}(1-u^2)^{-1/2}\), the law of conservation of momentum is now Lorentz covariant, making it a valid law in the framework of special relativity. Thirdly, let us inspect Newton’s theory of universal gravity. The basic equation in Newton’s theory of gravity is Poisson’s equation \(\nabla ^2\phi =4\pi \rho \), which indicates the relation between the gravitational potential \(\phi \) and the mass density \(\rho \).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In Chap. 6, we used \(\rho \) and \(\mu \) to represent the charge density and mass density, respectively. From this chapter on, since the charge density will show up less frequently, we will follow the convention of the majority and use \(\rho \) to represent the mass density.
- 2.
Up to now, we have been discussing in terms of Newtonian gravity. In Newton’s theory of gravity, there are two types of gravitational mass: active and passive. The former refers to the mass of an object as a source of its gravitational field, which determines the strength of the gravitational field it produces; the latter refers to the gravitational mass of the object as a test point mass in an external gravitational field, which determines the strength of the gravitational force it experiences in a given gravitational field. The gravitational mass in the main text refers to the passive gravitational mass.
- 3.
The gravitational field produced by the point mass is ignored (similar to the treatment of a test charge in electromagnetism).
- 4.
Principle (a) is put in the same way in all textbooks (although the formulation for the principle of general covariance may be different); however, there are at least two ways of stating the principle (b) in different books. The other one is: (b) the equivalence principle. With regard to the effects of the physical laws being derived, these two ways are equivalent. For details, see Sect. 7.5.
- 5.
We only discuss the case where \(G(\tau )\) is a non-self-intersecting curve, otherwise one will encounter causal difficulties (see Chap. 11 in Volume II). In fact, the timelike curves representing observers in this text are all assumed to be non-self-intersecting curves.
- 6.
Note that we are abusing the notation here, since \(\mathscr {F}_M(k,l)\) technically denotes the collection of all the tensor fields of type (k, l) on the manifold M but some fields in \(\mathscr {F}_G(k,l)\) here do not lie on the curve G.
- 7.
A non-gravitational experiment refers to an experiment where the gravitational interaction between the objects in the lab can be ignored, but there may exist a gravitational field produced by an object outside the lab (e.g., the Earth).
- 8.
However, this rule will lead to an ambiguity in the order of the operators when two derivative operators act successively; other considerations need to be taken into account to overcome this issue (See Sect. 7.2). Therefore, the claim that equivalence principles “can carry all the special relativistic laws of physics to curved spacetime” seems to be too strong. Misner et al. (1973) pp. 390–391 has a specific discussion on this.
- 9.
From the viewpoint of an observer on the Earth, the reason for A to bulge is the combination of two forces: (a) the Moon’s gravitational force, and (b) the centrifugal force caused by the Earth’s circular motion around the barycenter of the Earth and the Moon. The net force of these two forces is called the tide-generating force (or tide-raising force).
- 10.
A congruence of curves in U is a family of curves, such that for each \(p\in U\) there is a unique curve in this family passing through p.
- 11.
Some other conditions also need to be satisfied (e.g., non-self-intersecting).
- 12.
There exists such geodesic families in flat spacetime, in which we have \(u^b=0\) and \(a^c=0\) on a fiducial geodesic \(\gamma _0(\tau )\) (such as a parallel geodesic family). There also exists such geodesic families in flat spacetime, where we have \(u^b\ne 0\) on \(\gamma _0(\tau )\) [one can just let \(\gamma _0(\tau )\) and the nearby geodesic become not parallel]. However, there does not exist such a geodesic family, where \(a^c\ne 0\) on \(\gamma _0(\tau )\) unless the spacetime is not flat.
- 13.
- 14.
Specifically, the dependence of \(G_{\mu \nu }\) on the second order derivatives of \(g_{\mu \nu }\) is linear, while the dependence on the first order derivatives is quadratic. What is worse, \(G_{\mu \nu }\) also contains the inverse \(g^{\mu \nu }\) of \(g_{\mu \nu }\) (for raising the indices), which is very complex when expressed as a function of \(g_{\mu \nu }\).
- 15.
Conventionally, an electromagnetic field is not classified as a matter field, but as the source of a gravitational field we will later on refer it to as a matter field for convenience.
- 16.
In the linearized theory of gravity, people usually discuss the spacetime with the background manifold \(\mathbb {R}^4\), or a spacetime region where a flat Lorentzian metric \(\tilde{\eta }_{ab}\) can be defined. In the former case the Minkowski metric \(\eta _{ab}\) is globally defined, and in the latter case it is convenient to denote the (locally) flat metric \(\tilde{\eta }_{ab}\) as \(\eta _{ab}\).
- 17.
Also called the de Donder gauge condition or harmonic gauge condition of the linearized theory of gravity.
- 18.
Here we only require \(\eta _{ab}\) to be flat on U, and the same for Proposition 7.9.2 (see the first footnote in Sect. 7.8). It follows from Theorem 3.4.9 that for any (locally) flat metric there exists a coordinate system such that the metric components are constant. For Lorentzian signature, one can further find a coordinate transformation and turn them into \(\eta _{\mu \nu }\).
- 19.
Note that this is a handwaving discussion, since the constraint counting is actually very subtle when it comes to partial differential equations. For example, the second equation in (7.9.15) can be regarded as a constraint for \(\xi _0\) in the first equation, but it does not mean that \(\xi _0\) has no degree of freedom! For another example, the 1-dimensional wave equation \(\partial ^2_tu - c^{2} \partial ^2_xu = 0\) has the general solution \(u = f_{+}(x - ct) + f_{-}(x + ct)\), with \(f_{\pm }\) being arbitrary \(C^{2}\) functions of one variable. If the wave equation is considered to be a constraint, is the number of constraints 1 or \(-1\)?
- 20.
Notice that \(g_{ab} = \eta _{ab} + f(K_{\mu } x^{\mu }) H_{ab}\) is a pp-wave only in the linear approximation.
- 21.
In fact, this equation is \(R_{ac}=-(\partial _1\partial _1P+\partial _2\partial _2P)(e^4)_a(e^4)_c\). P taking the specific form in (7.9.64) makes \(\partial _1\partial _1P=f=-\partial _2\partial _2P\), which assures \(R_{ac}=0\).
- 22.
In fact, the metric for any pp-wave can be expressed in the Brinkmann coordinate system in the following general form:
$$\begin{aligned} \text {d} s^2=2P(u, x, y) \text {d} u^2-2 \text {d} u \text {d} v+\text {d} x^2+\text {d} y^2\,, \end{aligned}$$where P is an arbitrary smooth function. It is not difficult to see that this is equivalent to (7.9.65) (by setting \(v=\frac{t-z}{2}\)), and taking P to be of the form (7.9.64) is just a special case.
- 23.
According to Birkhoff’s theorem (see Sect. 8.3.3), the spherical evolution of any spherically symmetric star (such as collapse and oscillation) will not emit a gravitational wave no matter how dramatic it is, just like there does not exist a spherically symmetric electromagnetic wave in Maxwell’s theory. (The spherical wave of an oscillating electric dipole in a distant region is not a spherically symmetric electromagnetic wave, since the fields \({\vec {E}}\) and \({\vec {B}}\) are not spherically symmetric. In fact, a spherically symmetric electromagnetic wave corresponds to the radiation of an electric monopole, but this kind of radiation does not exist in Maxwell’s theory).
- 24.
More precisely, \(w^a\) only gives the direction of the “separation”, it is really \(w^a\Delta s\) (where \(\Delta s\) is small) that determines a neighboring observer in this direction, see Sect. 7.6.
References
Abbott, B. P. et al. (2016), ‘Observation of Gravitational Waves from a Binary Black Hole Merger’, Phys. Rev. Lett. 116(6), 061102. arXiv:1602.03837.
Cai, R.-G., Cao, Z., Guo, Z.-K., Wang, S.-J. and Yang, T. (2017), ‘The Gravitational-Wave Physics’, Natl. Sci. Rev. 4(5), 687–706. arXiv:1703.00187.
Carroll, S. M. (2019), Spacetime and Geometry, Cambridge University Press, Cambridge.
Chen, C.-M., Nester, J. M. and Ni, W.-T. (2017), ‘A brief history of gravitational wave research’, Chin. J. Phys. 55, 142–169. arXiv:1610.08803.
Fock, V. A. (1939), ‘Sur le mouvement des masses finies d’Apres la theorie de gravitation Einsteinienne’, J. Phys. U.S.S.R. 1, 81–166.
Geroch, R. P. and Jang, P. S. (1975), ‘Motion of a body in general relativity’, J. Math. Phys. 16, 65–67.
Hawking, S. W. and Ellis, G. F. R. (1973), The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge.
d’Inverno, R. A. (1992), Introducing Einstein’s Relativity, Clarendon Press, Oxford.
Liu, L. and Zhao, Z. (2004), General Relativity (in Chinese), Higher Education Press, Beijing.
Maggiore, M. (2018), Gravitational Waves: Volume 2: Astrophysics and Cosmology, Oxford University Press, Oxford.
Misner, C., Thorne, K. and Wheeler, J. (1973), Gravitation, W H Freeman and Company, San Francisco.
Ohanian, H. C. and Ruffini, R. (1994), Gravitation and Spacetime, W W Norton and Company, Inc., New York.
Sachs, R. K. and Wu, H. (1977), General Relativity for Mathematicians, Spinger-Verlag, New York.
Saulson, P. R. (2017), Fundamentals Of Interferometric Gravitational Wave Detectors, World Scientific, Singapore.
Stephani, H., Kramer, D., MacCallum, M. A. H., Hoenselaers, C. and Herlt, E. (2003), Exact Solutions of Einstein’s Field Equations, Cambridge University Press, Cambridge.
Straumann, N. (1984), General Relativity and Relativistic Astrophysics, Spinger-Verlag, Berlin.
Synge, J. L. (1960), Relativity: The General Theory, North-Holland Publishing Company, Amsterdam.
Wald, R. M. (1984), General Relativity, The University of Chicago Press, Chicago.
Weber, J. (1961), General Relativity and Gravitational Waves, Wiley-Interscience, New York.
Will, C. M. (1995), Stable clocks and general relativity, in ‘30th Rencontres de Moriond: Euroconferences: Dark Matter in Cosmology, Clocks and Tests of Fundamental Laws’, pp. 417–428. arXiv:gr-qc/9504017.
Will, C. M. (2014), ‘The confrontation between general relativity and experiment’, Living Reviews in Relativity 17(1), 4. arXiv:1403.7377.
Will, C. M. (2018), Theory and Experiment in Gravitational Physics, Cambridge University Press, Cambridge.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2023 Science Press
About this chapter
Cite this chapter
Liang, C., Zhou, B. (2023). Foundations of General Relativity. In: Differential Geometry and General Relativity. Graduate Texts in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-99-0022-0_7
Download citation
DOI: https://doi.org/10.1007/978-981-99-0022-0_7
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-99-0021-3
Online ISBN: 978-981-99-0022-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)