Abstract
Cosgrove demonstrates both that Minkowski spacetime taken as a physical concept is unnecessary to general relativity and that the mathematical apparatus of four-vectors is superfluous as well. He addresses the two principal respects in which Minkowski’s theory enters standard formulations of general theory of relativity: the field equation itself and the law of geodesic motion. Cosgrove concludes with the suggestion that the stress-energy tensor in the general field equation lacks physical intelligibility and might best be discarded in favor of the pure (“vacuum”) gravitational field equation.
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31 August 2018
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Notes
- 1.
Norton 1993, 799 –800.
- 2.
Page numbers otherwise unidentified in this section refer to Einstein’s 1916 paper “The Foundation of the General Theory of Relativity” (Einstein 1952b [1916]).
- 3.
The definitive discussion of the question is Norton 1989.
- 4.
Einstein 1952b [1916], 143.
- 5.
Einstein to Laue, September 12, 1950. Quoted in Norton 1989, 39–40.
- 6.
Synge 1960, ix.
- 7.
Stachel 1989a, 58.
- 8.
Einstein 1961 [1916], 109–110.
- 9.
- 10.
In a letter to Lorentz of January 1916 (Einstein 1998, 232–233), Einstein mentions three reasons for the downfall of the Entwurf theory: (1) it did not accommodate rotating frames of reference, (2) incorrect perihelion motion of Mercury, (3) the maximal degree of covariance he was able to achieve failed to uniquely determine the field equation.
- 11.
According to Norton (1987, 155), for instance, it is a “commonplace of differential geometry … [that] any well formulated spacetime theory is automatically expressible in coordinate free (= generally covariant) terms.”
- 12.
Gutfreund and Renn 2015, 183. Unfortunately , the introduction to the 1916 review article is omitted from the Perrett and Jeffery English translation. An updated English translation which includes the introduction is provided in Gutfreund and Renn, 183–232.
- 13.
Poincaré had already noticed it at least three years prior to Minkowski’s 1908 paper “Space and Time,” but Einstein does not appear to have been aware of Poincaré’s precedence in this respect.
- 14.
Einstein 1952b [1916 ], 119.
- 15.
Einstein 1953 [1922 ], 11.
- 16.
Einstein 1953 [1922 ], 12.
- 17.
Ryckman 2005, 150.
- 18.
Quoted in Gutfreund and Renn 2015, 81.
- 19.
Will 1993, chapter 6.
- 20.
For a metric geodesic in special relativity, the time component \( \frac{d^2t}{d{\tau}^2}=0 \)informs us merely that the relativistic factor γ is a constant, something that in any event follows necessarily from the space terms. If we then substitute \( \frac{dt}{\gamma }= d\tau \), we obtain \( \frac{d{x}^{\alpha }}{d{t}^2}=0. \)Clearly the time term \( \frac{d^2t}{d{t}^2}=0 \) is uninformative, with all the information regarding the trajectory of a free body contained in the three space terms.
- 21.
With regard to the “infinitesimal principle of equivalence,” there is no way of articulating a law of geodesic motion at all for an infinitesimal region of space and time since all trajectories, including non-geodesic ones, approach geodesic “straightness” in the infinitesimal. As Einstein writes to Moritz Schlick , based on the straightness of a trajectory in the infinitesimal “nothing can be derived…. [For] in the infinitesimal every continuous line is straight” (Heiraus kann aber nichts abgeleitet werden…. im Unendlichkleinen ist jede stetige Linie eine Gerade) (Einstein to Schlick , March 21, 1917; Einstein 1998, 417). Or in the more picturesque formulation of Torretti, to argue for a geodesic law of motion based on infinitesimal straightness “is like arguing that the parallels of latitude on the surface of the Earth are no less geodesic than the meridians, because they agree, for instance, with the avenues that cross Chicago from East to West, which are no less straight [in the infinitesimal] than those leading from North to South” (Torretti 1983, 316). Norton , 1989 (sections 9–10, 31–39), includes an informed discussion of this point.
- 22.
It is sometimes maintained that the restriction to a second rank tensor derives from the stress -energy tensor, which is of second rank and to which the gravitational tensor must be equated in the general field law. However, the restriction to second rank can be proved mathematically (Einstein 1922, 84) and does not depend on characteristics of the stress-energy tensor.
- 23.
Einstein 1982 [1954], 311. Similarly , in his “Autobiographical Reflections” of 1949 Einstein remarks, “The right-hand side is a formal condensation of all things whose comprehension in the sense of a field theory is still problematic. Not for a moment, of course, did I doubt that this formulation was merely a makeshift in order to give the general principle of relativity a preliminary closed-form expression” (Einstein 1979 [1949], 71).
- 24.
Arnowitt et al. (2008 [1962], 2024) treat the absence of pressure as a source term in the interior Schwarzschild solution with the remark that pressure is registered rather as “clothing of the original mechanical mass.” This suggests that pressure is subsumed by the energy density component of the stress -energy tensor and should not appear as a separate component in its own right.
- 25.
Vishwakarma 2012, 376.
- 26.
Vishwakarma 2012, 377.
- 27.
With c = 1, the four-displacement s is equal to the proper time τ, and so for small velocities s = t.
- 28.
\( {\varGamma}_{44}^{\tau }=\frac{1}{2}{g}^{\tau \alpha}\left(\frac{\partial {g}_{4\alpha }}{\partial t}+\frac{\partial {g}_{4\alpha }}{\partial t}-\frac{\partial {g}_{44}}{\partial {x}_{\alpha }}\right) \).
- 29.
Einstein 1952b [1916 ], 121.
- 30.
Einstein 1979 [1949 ], 71.
- 31.
Einstein 1953 [1922 ], 79.
- 32.
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Cosgrove, J.K. (2018). Minkowski Spacetime and General Relativity. In: Relativity without Spacetime. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-72631-1_7
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