Special Relativity in a Model Universe Michael S. A. Graziano Draft Aug 2009 Introduction The present paper describes a model universe called Hubs and Spokes (HS). HS operates on the basis of two rules. From those rules, a full description of HS can be generated. Objects can be constructed out of point particles according to a set of constraints. Below, the basic structure and rules of HS are defined. Then some examples of particle interactions in HS are described. Finally, the emergence of special relativity in HS is described. A peculiarity of HS is that special relativity and Galilean/Newtonian space-time are equivalent. For this reason, special relativity is particularly easy to understand in HS. Structure of HS 1. HS contains a space of three dimensions. These dimensions are parameterized here as x, y, and t. (The use of three dimensions instead of four is solely for the convenience of graphing the figures below. HS can be trivially extended to four dimensions.) These parameters form a reference frame termed R1. Other reference frames will be discussed below. 2. HS contains point particles. A point particle is a particle that is dimensionless in x and y but not in t. These particles are called grains here. Rules of HS 1. When a grain is not interacting with another grain, it moves in a straight line at speed c with respect to reference frame R1. Thus a non-interacting grain follows the equation: [Eq.1] Δx 2 + Δy 2 - c 2 Δt 2 = 0. This rule is expressed in a frame-dependant manner, as if R1 were a preferred coordinate frame. Later it will be shown that Rule 1 is also valid in other reference frames. 2. Grains interact by absorption or emission. An absorption is when two grains combine on contact to produce a single resultant grain. An emission is a time-reversed absorption. 1
Some examples Consider two grains, 1 and 2, traveling on paths that converge on a point in space A at time t. No rule requires an absorption to occur. The grains might pass through each other and continue their separate paths. If an absorption does occur, by Rule 2, the absorption requires contact between the two grains and thus can occur only at point A. In this scenario, grains 1 and 2 combine to produce a single resultant grain 3. This interaction is shown in Figure 1, which illustrates the paths traced by the grains in the x and y dimensions. y grain 3 A grain 1 grain 2 x [Figure 1: Grains 1 and 2 converge at point A; an absorption occurs on contact; grain 3 results. Arrows indicate the direction of movement along the paths of the grains.] By Rule 1, grains 1 and 2 approach each other on straight paths at speed c and grain 3 leaves the point of interaction on a straight path at speed c. The rules of HS do not specify the angles between these three paths; any angles are permissible. There is no such thing as momentum in HS, and no conservation laws govern the resultant trajectory in this interaction. Figure 2 shows two alternative ways to conceptualize the same situation. In the first way (Figure 2, left), grain 1 absorbs grain 2 and then continues its trajectory while grain 2 no longer exists. In this conceptualization, grain 1 can change its direction of motion at the point of absorption. This change in direction is allowed because Rule 1, which specifies a straight path, 2
applies only to grains while they are not interacting and thus does not apply to point A, the point at which the interaction occurs. [Figure 2: Two different ways to label the events in Figure 1.] In the second way to conceptualize the situation (Figure 2, right), grain 2 absorbs grain 1 and then continues its trajectory while grain 1 no longer exists. These different ways to conceptualize the interaction between grains are equivalent and merely represent different methods of labeling. They are important, however, in that they illustrate a fundamental property of HS: A grain can change its direction of motion only during an absorption or emission event. d > 0 d =0 y grain 5 y grain 4 B grain 5 grain 3 d grain 4 A grain 1 grain 2 grain 1 grain 2 x x [Figure 3: Two more allowable constructions in HS.] 3
Figure 3 (left) shows another construction allowable in HS. Grains 1 and 2 converge at point A and absorb on contact to produce grain 3; grain 3 travels from A to B; an emission occurs at B, resulting in grains 4 and 5 leaving point B. Let d = the distance between point A and B. Since d can be any arbitrary length, consider the limit as d approaches zero, illustrated in Figure 3 (right). In this construction, two grains converge and two grains leave the point of convergence. Using similar constructions, it is easily shown that any whole number of grains >2 can be joined together at a single vertex. The only constraint is that the grains travel toward and away from the vertex in straight lines at speed c. In this sense, HS is a tinker-toy universe in which straight line-segments can be jointed together at vertices. Rule 1 supplies the line segments; Rule 2 supplies the connecting hubs. It is this interaction between the two rules that provides the spatial structure of HS, as described in later sections. Consider the construction in Figure 4. Grain 1 undergoes an emission event at point A, resulting in grains 2 and 3. Let θ = the angle between the paths of grains 2 and 3. The rules of HS do not constrain this angle; any value is permissible. y 2! 3 A 1 x [Figure 4: grain 1 travels to point A; an emission occurs; grains 2 and 3 leave point A.] 4
In the special case in which θ = 0, grains 2 and 3 follow the same path and thus are in constant contact. In this case, an absorption can occur, resulting in a single grain 4. This case is diagrammed in Figure 5 (left). Here the construction looks as though a grain is changing its direction of motion spontaneously, but note that the changes of direction still occur at absorption or emission events. Now let point A be brought infinitesimally close to point B; and let grain 4 be re-labeled as grain 1. Figure 5 (right) shows this specialized case in which grain 1 can change its direction due to an emission event followed an infinitesimal time later by an absorption event. This example shows that a vertex in HS can be a joining of as few as two lines. The change of direction of grain 1 seems peculiar, violating expectations of what is permissible in the real universe. We are used to a universe in which momentum is conserved. However, momentum is not defined in HS and the change of direction of grain 1, due to an emission event followed an infinitesimal time later by an absorption event, is permissible. y y 2 & 3,! = 0 A B 4 1 1 1 x x [Figure 5: Permissible constructions in HS. Left: grain 1 emits grains 2 and 3 at point A. Grains 2 and 3 follow the same path and absorb each other at point B to produce grain 4. Right: the same construction but with points A and B infinitesimally close to each other and grain 4 relabeled as grain 1.] 5
Stationary objects in HS Thus far we have considered the behavior of grains in HS that travel at speed c in reference frame R1. It is possible to construct objects in HS that are stationary in R1. Figure 6 shows an especially simple example. In the first time period (Figure 6, left), grain 1 travels from point D to point A and grain 2 travels from point B to point A. At A, grain 1 absorbs grain 2 and then an infinitesimal time later emits it. During the second time period (Figure 6, center), grain 1 travels from A to D, while grain 2 travels from A to B. Two other grains, 3 and 4, follow similar motions along opposite sides of the construction. These four grains interact at the four vertices and continually bounce back and forth, forming a construction that is stationary in R1. Note that there is nothing to hold the grains to these trajectories. It is equally permissible in HS that the grains, upon absorption and re-emission, fly off at odd angles and never encounter one another again. The object in Figure 6 is only one arbitrary member of the infinite set of permissible constructions in HS. Because there are no dynamical laws in HS, there is no way to assess the likelihood that the object in Figure 6 will actually occur in HS. All we can do is take the tinker-toy components given to us and explore what objects can and can t be built out of them. The object in Figure 6 can be built, and belongs to the specific subset of stationary objects. [Figure 6: An example of a stationary object permissible in HS. The pattern of motion of this object repeats indefinitely.] Since the size of the object in Figure 6 is arbitrary, consider the case in which the height and width approach zero. The object in this case becomes dimensionless in x and y. It is 6
reduced to a point particle. Thus, even though grains are constrained to travel at speed c with respect to R1, it is possible to construct a point particle that is stationary in R1. Indeed, it is possible to construct objects and point particles in HS that move at any speed v c. It is not possible to construct objects in HS that move at speeds greater than c, because the component grains of all objects are fixed at speed c. Moving objects in HS Consider the object in Figure 7. (All features of the object in Figure 7 are designated with an underline.) It is constructed to be similar to the object in Figure 6. It has the same number of grains interacting at the same number of vertices. Segments AB and DC are parallel to the x axis, and segments AD and BC are parallel to the y axis, matching the object in Figure 6. Its height, AD, is equal to the height of the object in Figure 6, AD. However, the object in Figure 7 is moving in the positive x direction at some speed v < c. That is, points A, B, C, and D are all moving to the right at speed v. Because of this motion, the object in Figure 7 must have a different structure from the object in Figure 6. Three main differences are discussed below. [Figure 7: This object is moving to the right at speed v in R1.] Clock rate In the object in Figure 6, grain 1 travels from A to D and back to A. Its pattern of motion repeats after this round trip, thus after a time interval of: [Eq.2] "t 1 = (2AD). c 7
In the object in Figure 7, grain 1 travels from A to D and back to A, traversing the height of the object. At the same time, the grain is moving to the right at speed v, in tandem with the object. Since the total speed of the grain is fixed at c, and the horizontal speed is v, the vertical speed of the grain must be (c 2 -v 2 ) 1/2. The pattern of motion of grain 1 therefore repeats after a time interval of: [Eq.3] "t 1 = (2AD) (c 2 # v 2 ) 1/ 2. Since we have set the two objects to the same height: [Eq.4] AD = AD. Combining equations 2-4, we find: [Eq.5] "t 1 = "t 1 (1# v 2 /c 2 ) 1/ 2. Thus, grain 1 repeats its pattern of motion on a slower time scale than grain 1. That is, the rate at which internal events occur for the moving object in Figure 7 is slower than for the stationary object in Figure 6. The greater the speed v, the longer the time interval Δt between these internal events. In the limit as v approaches c, all internal motion in the object will stop. The component grains will be constrained to move in the positive x direction in straight lines at speed c. Length For the object in Figure 6, we know that grain 1 and grain 2 must always reach point A at the same time for the required absorption to occur. Therefore, the round-trip time for grain 1 must equal the round-trip time for grain 2. Thus: 8
2AD/c = 2AB/c, or: [Eq.6] AD = AB. That is, the height equals the length. For the object in Figure 7, again, grain 1 and grain 2 must always reach point A at the same time. Therefore the round-trip time for grain 1 must equal the round-trip time for grain 2. Thus: [Eq.7] 2AD AB = (c 2 " v 2 1/ 2 ) (c " v) + AB (c + v). Combining equations 4, 6, and 7: [Eq.8] AB = AB(1" v 2 /c 2 ) 1/ 2. That is, the length of the object in Figure 7 does not equal the length of the object in Figure 6. Instead, the length in Figure 7 is shortened. In the limit as v approaches c, the length of the object approaches zero. Simultaneity of events Consider grains 1 and 2 in the object in Figure 6. The grains leave point A at the same time (Figure 6, center). They travel the same distance at the same speed, thus grain 1 reaches point D at the same time that grain 2 reaches point B. The absorption that occurs at point D, and the absorption that occurs at point B, are therefore simultaneous. For the object in Figure 7, the two corresponding events are not simultaneous. Grains 1 and 2 leave point A at the same time. Grain 1 arrives at point D after a time AD/(c 2 -v 2 ) 1/2. Grain 2, however, arrives at point B after a time AB/(c-v). The time difference between these two events is as follows: 9
[Eq.9] t (event at B) - t (event at D) = AB (c " v) " AD (c 2 " v 2 ). 1/ 2 By using equations 4, 6, and 8, we can express the right-hand side of equation 9 in terms of AB: [Eq.10] t (event at B) - t (event at D) = AB(v /c 2 ) (1" v 2 /c 2 ) 1/ 2. That is, events that were simultaneous in the stationary object in Figure 6 are not simultaneous in the moving object in Figure 7. The time difference between the two events in Figure 7 is proportional to the x distance between the corresponding two events in Figure 6. The object in Figure 6 and the object in Figure 7 are therefore clearly different. We can try to construct them in the same way, with the same number of grains interacting at the same number of vertices. We can align their segments in the same way with respect to the x and y axes. We can give them the same height. Beyond these similarities, however, the rules of HS require the objects to be different. The object that is moving in R1 repeats on a slower time scale; its length is shortened along the direction of motion; and events that were simultaneous in the stationary object may not be simultaneous in the moving object. Moving reference frames in HS. Consider a reference frame other than R1. Let R2 be a reference frame moving in the positive x direction at speed v with respect to R1. For sake of ease of explanation, let us accept the somewhat fanciful supposition that HS is inhabited, and the creatures in R1 and R2 are in communication. They attempt to define a set of measurement standards so that they can communicate in a meaningful fashion. R1 chooses a standard object; for example, the square object in Figure 6. The height and length of this object provide standard units of distance in the x and y dimensions. Each vertex of the object is the location of an event, an emission of one grain by another, that repeats in time. The time between these repeating events can serve as a clock. Let us call this object S1 for the standard object in R1. It is like a pocket ruler and 10
stop-watch that R1 can use to measure events. By tiling the space of HS with many copies of S1, the observer R1 can construct a coordinate grid. Now consider an object S2, a similar standard object but at rest in R2. In order to function as a meaningful measurement standard, S2 must be calibrated with respect to S1. But we have already seen that S1 and S2 are required to be different in some respects. Thus observer 1, using S1 to build a coordinate grid, may measure an event to occur at coordinates x, y, and t; whereas observer 2 using S2 to build a coordinate grid will arrive at a different set of measurements, x, y, and t. Suppose the coordinate grid built of S1 and the coordinate grid built of S2 are aligned such that both have the same origin. The point (x=0, y=0, t=0) is the same as the point (x =0, y =0, t =0). Then the equations that relate the primed and unprimed variables can be derived with a small amount of algebra using the equations that relate S1 and S2. They are: [Eq.11] [Eq.12] x'= y'= y x " vt (1" v 2 /c 2 ) 1/ 2 [Eq.13] t'= t " xv /c 2 (1" v 2 /c 2 ) 1/ 2 These equations are the Lorentz transformation equations of special relativity. They were derived using the objects S1 and S2 as measurement standards. However, any object of sufficient complexity that is stationary in R1, and its matching counterpart that is stationary in R2, can be used to arrive at the same transformation equations. (To step through this derivation in the general case of an arbitrary object is somewhat tedious and is not included here for the sake of brevity.) The transformation equations are independent of how the measuring rods and clocks are constructed. They are a consequence of the rules of HS. 11
Special Relativity in HS Special relativity was originally derived by Einstein from two postulates: the constancy of the speed of light, and the principle of relativity, or the principle that the laws of physics are the same for any observer in constant motion. Do these two principles hold true in HS? Constancy of the speed of light. Consider a grain moving from one point to another. R1 measures the locations and times and calculates the speed of the grain as c (by definition, according to Rule 1). R2 also measures the same events and calculates the speed of the grain. Equations 11 13 imply that R2 will also measure a speed of c. That is, the speed of any grain is always measured as c regardless of the reference frame. Thus the first postulate of special relativity, the constancy of the speed of light in all reference frames, holds true in HS. Principle of relativity. Rule 1 of HS states that when grains are not interacting they travel at a fixed speed c in reference frame R1. We now know that this rule also holds true in reference frame R2. Rule 2 states that grains can absorb or emit each other. This rule is independent of any specific reference frame. Thus, both rules of HS are the same in any arbitrary reference frame in uniform motion, satisfying the postulate of relativity. Note that both postulates of special relativity had to be derived in HS. They were not provided at the outset as postulates. In particular, Rule 1 is expressed in a frame dependant fashion, as if reference frame R1 were a preferred reference frame. It is only by examining the properties of HS that we find out that R1 is not a preferred reference frame and that Rule 1 is unchanged in other reference frames. Einstein vs Newton Perhaps the most peculiar property of HS is that we can choose between an Einsteinian view or a Newtonian view. Both views are correct. From a Newtonian perspective, we can arbitrarily declare one reference frame (for example R1) to be at absolute rest. The measurement discrepancies between that rest frame and other, moving reference frames can be understood in terms of the mechanical combining and splitting of particles that move at constant speed through a Newtonian space-time. For example, a clock translating through space at high speed has its inner workings slowed down. The reason 12
is that its components are fixed at a set speed. If part of that speed is already taken up in the global, translational motion of the clock, then there is less speed left over for the inner workings of the clock. In the limit as the clock reaches the speed c, its components are already translating at their full speed and cannot move with respect to each other, hence all internal motion in the clock must stop. From an Einsteinian perspective, the rules of HS are the same in all reference frames; no experiment can distinguish one reference frame as an absolute rest frame; and space and time are intermixed. Both descriptions, the Newtonian and the Einsteinian, provide mathematically accurate descriptions of HS. They are equivalent. 13