Abstract
This chapter is a critical reading of the published version of Minkowski’s 1908 Cologne address, in which the concept of four-dimensional spacetime was publically unveiled. Cosgrove places Minkowski’s endeavors in relativity theory in the context of formalism-driven “Göttingen science” of the early twentieth century. Turning to Minkowski’s four-dimensional kinematics, Cosgrove sees the latter as simply a formal-mathematical representation of Einstein’s 1905 special relativity with no physical coherence or additional physical insight beyond Einstein’s theory. Cosgrove finds Minkowski’s four-dimensional vector formulation of relativistic mechanics to be once again merely of formal-mathematical significance with no physical intelligibility.
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31 August 2018
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Notes
- 1.
- 2.
Corry 1997, 274–275. Such focus on logical consistency is indeed exemplified by Minkowski’s address, as when, after introducing his “world postulate,” Minkowski sets out to demonstrate that “the assumption of the group G c for the laws of physics never leads to a contradiction ….” (Minkowski 1952 [1909], 86).
- 3.
Quoted in Pyenson 1977, 89.
- 4.
On Göttingen science in general during the period in question, see Heelan 1987, sec. 2, 371–373.
- 5.
See Lehmkuhl 2014.
- 6.
Minkowski 1952 [1909 ], 75.
- 7.
Damour 2008, 626 –628.
- 8.
Damour 2008, 629 .
- 9.
Friedman 1983, 306 .
- 10.
Minkowski 1952 [1909 ], 83.
- 11.
- 12.
Various authors have noted Minkowski’s failure in the Cologne lecture to mention Poincaré’s contributions. Since it is Minkowski’s formulation that was taken up into subsequent history of mathematical physics, for our purposes we can regard Minkowski as the “inventor” of four-dimensional spacetime. On Poincaré’s contributions to special relativity in relation to Minkowski see, for instance , Damour (2008), section II, and Walter (1999), section 2.2.
- 13.
A hyperbolic rotation sweeps out angles on a hyperbola rather than on a circle as in regular trigonometry. In “Space and Time,” Minkowski declines to use the hyperbolic functions sinh and cosh, although presumably he knew how to employ them. The hyperbolic functions leave the minus signs for the spatial variables intact and are in that respect less hospitable to the desired formal analogy with the Pythagorean Theorem .
- 14.
Presumably a finite value is more intelligible because if c were infinitely large, then \( \frac{dx}{dt} \) would reduce to \( \frac{dx}{0} \). For an infinitely large velocity, that is, no time elapses during the traversal of any distance.
- 15.
See, for instance , Mermin 1984.
- 16.
Eddington 1965 [1923 ], 10.
- 17.
Eddington actually writes for the time component dy 4 2, but for consistency I will adopt dx 4 2.
- 18.
Friedman 1983, 138 –142.
- 19.
Friedman 1983, 138 –139.
- 20.
Friedman 1983, 142 .
- 21.
Friedman is not explicit that the structure of Minkowski spacetime is the only sure way to the Lorentz transformation , as opposed to simply the best way, although that is certainly the impression I get.
- 22.
The prominence of Klein’s program in Minkowski’s approach to relativity is somewhat downplayed by Scott Walter (Walter 2014).
- 23.
Einstein’s definition of the length of a moving rod as the distance between the simultaneous coordinates of its endpoints essentially defines an instantaneous relative space.
- 24.
Einstein 1952a [1905], 49–50.
- 25.
The derivation can be found in any beginning textbook on relativity. Since the value of the four-velocity is c, it follows that u 2 = c 2. If we then differentiate with respect to τ, we obtain using the chain rule 2(u ∙ d u/dτ) = 0 or u ∙ a = 0, and so u ∙ f = 0. As a rule, I shall identify four-vectors by boldface type.
- 26.
Schutz 1985, 44 .
- 27.
Nerlich (2013, 188–122) argues that the proper time should be understood as a frame-independent quantity rather than as a quantity relative to any particular reference frame (or all of them). I shall address the distinction between frame invariance and frame independence in Chap. 3 below. Here I simply observe that even on Nerlich’s interpretation the four-velocity is a hybrid construction, since we take the derivative of a relativistic quantity (coordinate time) with respect to an “absolute” quantity (proper time).
- 28.
Einstein 1953 [1922 ], 11.
- 29.
Newton 1999 [1726], 416.
- 30.
Newton 1999 [1726], 405.
- 31.
Newton does not similarly distinguish between a qualitative definition of “quantity of motion” and the quantification of the same in a law of motion: “Quantity of motion is the measure of motion that arises from the velocity and the quantity of matter jointly” (Definition 2 of Newton 1999 [1726], 404). Presumably, the omission is because “quantity of motion” is already the quantification of the qualitative concept of motion. What would rather require qualitative definition would be motion itself, a concept Newton presumably takes as primitive and therefore indefinable.
- 32.
Newton typically speaks of single quantities being “proportional” to other quantities, but obviously the proportionality of ratios is intended, since a proportion equates ratios, not quantities. Niccolò Guicciardini argues that Newton’s practice of referring to “proportions” between single quantities at specific points (as for instance Prop. VI, Book I of the Principia) suggests that Newton actually derived these results by means of algebra . See Guicciardini 1999, 125–135. Today we express a proportion algebraically by means of a “constant of proportionality.”
- 33.
In the appendix (“Mechanics and the relativity postulate”) to Minkowski’s earlier and more technical paper on relativity (Minkowski 2012b [1908], 51–110) there is no such detour through relativistic force. One must presume that in “Space and Time” Minkowski is proceeding for the benefit of physicists unfamiliar with the four-vector calculus.
- 34.
I have corrected the Perrett and Jeffery translation, where “divided by c 2” in the quoted passage incorrectly reads, “divided by \( c \).” The Lewertoff and Petkov translation (Minkowski 2012c) of “Space and Time” corrects this typographical error. I have also substituted γ for Minkowski’s \( \frac{d t}{d\tau} \) and v x for his \( \raisebox{1ex}{$\frac{d x}{d\tau}$}\!\left/ \!\raisebox{-1ex}{$\frac{d t}{d\tau}$}\right. \) (and likewise for the other spatial variables).
- 35.
(cf t , Fγ) ∙ (cγ, vγ) = 0, and so c 2 f t γ − Fvγ 2 = 0 or \( {f}_t=\frac{Fv\gamma}{c^2} \). Once again, F is the relativistic force.
- 36.
This follows from the fact that \( {v}_xX+{v}_yY+{v}_zZ=\frac{d{E}_k}{dt} \) by the work-energy theorem.
- 37.
Hartle 2003, 87 .
- 38.
Nerlich 2013, 64 .
- 39.
Petkov 2005, 113 .
- 40.
Even Einstein himself, uncharacteristically, seems to have fallen prey to the temptation. In his Autobiographical Notes, for instance, Einstein remarks that if “one introduces as the unit of time, instead of the second, the time in which light travels 1 cm, c no longer occurs in the equations. In this sense one could say that the constant c is only an apparent universal constant. It is obvious and generally accepted that one could eliminate two more universal constants from physics by introducing, instead of the gram and the centimeter, properly chosen “natural” units (for example, mass and radius of the electron). If one considers this done, then only ‘dimensionless’ constants could occur in the basic equations of physics ” (Einstein 1979 [1949], 59). This is all clearly false and represents a seduction by pure mathematics to which Einstein was not normally susceptible. An adjustment of units does not remove c from the equations in any sense save a notational one. Clearly, c still maintains its dimensionality even if adjusted to “one unit of distance per one unit of time.” It is hard to imagine Einstein really thinking otherwise.
- 41.
Nerlich 2013, 91 .
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Cosgrove, J.K. (2018). Minkowski’s “Space and Time”. In: Relativity without Spacetime. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-72631-1_2
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