In his book chapter, Bourne introduces the layman reader to the basic non-mathematical postulates of special relativity. He begins by presenting our common-sense intuitions about the additivity of speeds between objects, which turns out to be violated, since no speed can be added to exceed the speed of light c. This has some counterintuitive consequences for concepts like simultaneity and the uniform passage of time.
It is worth to start by putting Bourne’s chapter into the context of his entire book. His overall project is to defend a thesis in the philosophy of time known as presentism, which postulates that only present entities exist, as opposed to eternalism, or the thesis that all past, present and future are ontologically tantamount. His particular take on this view is called “ersatzer presentism” and posits that times are abstract entities that are real, yet only the present is concretely realised. One of the greatest challenges to this theory comes from special relativity, hence its introduction in chapter 5. Specifically, the “Putnam-Stein debate” evolved around the idea that in the light of failure of simultaneity in special relativity, tensed propositions did not refer to any real physical concept. How Bourne overcomes these challenges is beyond a comprehension exercise, but suffice it to say that he joins the Neo-Lorentzians in arguing that an absolute present moment for the universe can be preserved under relativity.
Let us now turn to the specific points he picks out from the Einsteinian theory in the relevant chapter. Firstly, for any event, there are different frames of reference, from which that event can be observed. Imagine a skateboarder (I apologise for the quality, I drew and scanned these and formatted the formulas myself):
In picture (i) a person throws a ball which from his point of view or frame of reference and ours moves at v1. In (ii) the same occurs on a skateboard. From our frame of reference the ball moves at v1 (its velocity in the frame of the skateboarder) + v2 (the velocity of the system from our frame). This additivity is governed by the Galilean transformations. Let us consider a person with a torch.
The light from the torch moves in a straight line according to the “law of the propagation of light” at c, hence we would expect the combined velocity of light rays in picture (iv) to be v1+c, which is false. Instead we follow the “limit principle”, which posits that c is the universal speed limit (“if this red sign looks blue, then you are driving too fast”), which cannot be increased by moving with it. Einstein’s great intuition in solving this dilemma was to assert if c is constant and a function of space and time, and the space variable is allowed to chance, then the time variable cannot be a constant. Instead, to ensure c, time moves slower in the frame of reference of the skateboarder to offset his movement in space. For Bourne the prima facie philosophical implication is that, if duration is not invariant, then it cannot be an objective feature of reality.
By how much time slows down or “dilates” for the skateboarder is governed by the Lorentz-transformations that introduce the relativistic factor gamma.
Gamma is inverse-exponentially related to the difference between the speed of an entity and c, such that as the skateboarder skilfully approaches c, 𝛾 reaches infinity as shown on the graph above. This factor can be plugged into the observed time of the skateboarder where to is his time passage at rest, and t is time adjusted by his movement. The same occurs to his mass and inversely his length as he travels.
Let us consider an example hereof. The solar system travels at about 200km/s within our galaxy. If light is sent from the sun to either a mirrored planet in the path this travel, or adjacent to it and then reflected, light ought to return simultaneously. To ensure this, the solar system becomes elliptic and contracts in length, such that the distances offset the extra time required. Other consequences include that the skateboarder will age slower than the us and be heavier during his travels.
However, more interesting than the transformations are what does not change, namely the metric for calculating the objective distance between two events in space-time as space and time offset each other. Consider a coordinate system containing a line that, if the system moves changes in x and y size while objectively staying the same size L under Pythagoras.
Similarly, space-time with its three spacial axes and a timeline together as coordinate system will contain points separated by some objective distance regardless of changes in either of the axes. This distance can be calculated by the flat relativistic metric:
It ignores space-time curvature and is hence only accurate in some regions of our universe, but the basic idea of invariance and that “space by itself, and time by itself need to be rejected as entities in their own right” remains (for a general measurement, see the Schwarzschild metric), and their measurement is dependent on one another. For Bourne, the negativity of the space interval -c2(𝚫t)2 implies that time and space are of fundamentally different character in their mathematical treatment, which is a welcomed outcome for presentism.
The chapter goes on to flesh out the notion of a tense under this picture with the use of Minkowskian space-time cones that show the all coordinates an entity can occupy in its past and future given c as speed limit, all of which I shall not discuss now. Quintessentially, Bourne uses the idea to arrive at a notion of absolute past and future, which reside in different areas of the cones. This in turn he hopes will help in defending presentism by showing how the present is an objective feature of relativity. Whether he succeeds however, shall be discussed in an actual essay. Let me instead briefly turn to the very elegant result of the Lorentz-transformations. Consider the orbital speed of earth put into gamma:
Gamma is just larger than 1, but significantly so. For such small numbers, gamma can be expanded into the “post-newtonian approximation” like so:
where the blue fractions are so small that they are regarded as negligible. We can now include gamma in the transformation for mass of the earth.
Note that the red bit is the formula for kinetic energy, while M0 is an expression for the earth’s mass’ rest energy. Hence we have the ingredients to formulate the famous proposition of E=mc2, which expresses more than special relativity intended, namely the similarity and interchangeability of mass and energy. This makes the theory now incredibly neat and tidy: If (v2/c2) is close to 0, then Galilean transformations hold; if (v2/c2)=1, then we find that c is the limit; if (v2/c2)=ε, then E=mc2 (where ε is the symbol for a very small yet significant number. As illustration, the mathematician Erdos called his children epsilons).
Lastly, the very part of special relativity that forbids backward time travel is the limitation of c. However, the hypothetic backward time-travelling Tachyon particle often used in Star Trek is a mathematical example of what were to happen if v>c, namely 𝛾 would be:
Hence with 𝛾, t would be to divided by an imaginary number.
Bailyn, C. 2007. Frontiers and Controversies in Astrophysics. Yale Open Courses. Lectures 9-11. URL=[oyc.yale.edu/astronomy/astr-160/lecture-9]. accessed 01.09.2010.
Balashov, Y. 2007. Review of A Future for Presentism. Notre Dame Philosophical Review Journal. URL=[ndpr.nd.edu/news/23018-a-future-for-presentism]. accessed 31.01.2013.
Bourne, C. 2006. A Future for Presentism. Oxford.