Abstract
Rohatinski begins his analysis by examining how time has been viewed historically as an objective category. Starting with Aristotle’s notion of time in motion, the author continues the survey through Isaac Newton’s perspective of time as absolute, true, and mathematical, unaffected by the universe or the processes within it. Further, Rohatinski explores how, beginning in eighteenth century, physicists began questioning whether time was actually an exogenous force, a line of enquiry that led to Albert Einstein’s hypotheses about the relativity of time.
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Notes
- 1.
Isaac Newton , Philosophiae Naturalis Principia Mathematica, Bk. 1 (1689); trans. Andrew Motte (1729), rev. Florian Cajori (Berkeley: University of California Press, 1934) 6.
- 2.
Ibid., 7–8.
- 3.
Commenting on Newton’s theory, Einstein concluded that absolute time and space do not imply only “physically real,” but also “independent in their physical properties, containing physical elements, but not (space and time) affected by physical conditions.” Albert Einstein, The Meaning Of Relativity, 3rd ed. (New Jersey: Princeton University Press, 1950).
- 4.
Roger Joseph Boscovich, A Theory of Natural Philosophy (Chicago and London: Open Court, 1922). According to Boscovich, real space and time cannot be directly cognised by using senses, and the local modes of existence cannot be distinguished in time and space because “our ideas” or perceptions always showed the difference between the new state and the previous one, and not an absolute change, which is not within reach of our senses.
- 5.
Ibid., 407.
- 6.
The relative velocity is not negligible in relation to the speed of light .
- 7.
As presented in V.A. Ugarov, Special Theory of Relativity (Moscow: Mir Publishers, 1979) 126–128. In Euclid’s space , for example, in Descartes’ coordinate system, the square of the distance between the two points is determined by the Pythagorean theorem according to which:
$$ {\displaystyle \begin{array}{l}\kern7.5em \mathrm{d}{s}^2=\mathrm{d}{x}^2+\mathrm{d}{y}^2+\mathrm{d}{z}^2\kern0.28em \hfill \\ {}\left(\mathrm{where}\ \mathrm{d}{x}^2={\left({x}_2-{x}_1\right)}^2;\mathrm{d}{y}^2={\left({y}_2-{y}_1\right)}^2;\mathrm{d}{z}^2={\left({z}_2-{z}_1\right)}^2\right)\hfill \end{array}} $$In Minkowski’s system:
$$ \mathrm{d}{s}^2={\tau}^2-\mathrm{d}{x}^2-\mathrm{d}{y}^2-\mathrm{d}{z}^2\kern0.84em \left(\mathrm{where}\ \tau = ct\right) $$In such a system there is a uniform manner to establish the temporal and spatial distance between events, which both appear as definitive points each with a specific position in space and an instant in time. Here, the temporal-spatial succession constitutes a “world line” of an object which the events refer to and which is (viewed in the coordinate system x, τ) parallel with the “temporal” τ axis if an object is located at a fixed point on the spatial x axis, that is forming with it an angle Θ (Θ = arctan v/c), if the object moves along the x axis at the constant velocity v. If the event in relation to which we measure the temporal-spatial distance of other events is assumed to be “located” at the origin of the coordinate system (x 1 = 0, τ 1 = 0), any other event taking place on the x axis before or after the first event will be visualised by a point in the system (x, τ), with the square of the distance in relation to the origin (the first event) expressed by the following definition
$$ {s}_{1,2}^2={\left({\tau}_2-{\tau}_1\right)}^2-{\left({x}_2-{x}_1\right)}^2={\tau}^2-{x}^2. $$Straight lines (world lines of light beams) satisfying the condition, τ 2 − x 2 = 0, that is τ = x(ct = xi − ct = x), and on which the distance between event—by definition—equals zero, divide the system in four quadrants. Quadrants I and II signify the temporal distance of the certain event located in the origin, while quadrants III and IV signify their spatial distance. Thereby, since in the first quadrant s 2 > 0 and τ > 0, the events located in it absolutely follow after the event located in the origin; that is, it signifies an absolute future, unlike the second quadrant where s 2 > 0 but τ < 0 which thus indicates an absolute past. In contrast, in the third and the fourth quadrants s 2 < 0, and as a result, they signify an absolute spatial distance of the event (from the event located in the origin), provided that this event might take place before, after, or simultaneously with the event located at the origin. So the terms “simultaneously,” “before,” and “after” are relative for the events located in these quadrants.
Apart from this, in every quadrant there is an equilateral hyperbola which signifies the equidistant points in relation to the origin of the reference frame , with a shape defined by the relation of s 2 = τ 2 − x 2, so that in the reference frame (x, τ) there are two pairs of equilateral hyperbolas: τ 2 − x 2 = 1 (in quadrants I and II) and x 2 − τ 2 = 1 (in quadrants III and IV).
Relative simultaneity of events, time dilation between the inertial reference frames moving relative to each other at the relative velocity v, in Minkowski’s four-dimensional temporal-spatial system is expressed as:
-
All the events in the reference frame K with the “world line” parallel with x axis (i.e. for which τ = constant) are simultaneous, the same as all the events in the reference frame K′, with the “world line” parallel with x′ axis (i.e. for which τ′ = constant) are also simultaneous.
-
Relative motion of the two reference frames causes, in accordance with the Lorentz transformations , an inclination between the axis τ and τ′ and axes x i x′ (in relation to the common origin of these systems in which τ = τ′ = 0 and x = x′ = 0), the size of which is determined by the angle Θ = arctan v/c.
-
Distortion of parallelism of the coordinate axis between frames K and K′ destroys the absolute simultaneity of the events viewed from these systems, thus causing the time dilation (and, by analogy, the length contraction):
$$ \varDelta t=\frac{\varDelta {t}_0}{\sqrt{1-{v}^2/{c}^2}} $$ -
- 8.
For Whitehead duration meant “repetition of a given pattern in successive events.”
- 9.
Alfred North Whitehead, Science and the Modern World (New York: Macmillan Co., 1925) 125.
- 10.
Ibid., 125.
- 11.
Ibid., 159.
References
Aristotle. 1952. Physics. New York: Oxford University Press.
Boscovich, Roger Joseph. 1922. A Theory of Natural Philosophy. Chicago and London: Open Court.
Einstein, Albert. 1950. The Meaning of Relativity. New Jersey: Princeton University Press.
Newton, Isaac. 1962. Philosophiae Naturalis Principia Mathematica. Berkeley: University of California Press.
Platon. 1999. Timaeus. Newburyport, MA: Focus Publishing.
Ugarov, Vladimir A. 1979. Special Theory of Relativity. Moscow: Mir Publishers.
Whitehead, Alfred North. 1925. Science and the Modern World. New York: Macmillan Co.
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Rohatinski, Ž. (2017). Time as an Objective Category. In: Time and Economics. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-61705-3_2
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