Rivers of Time - Theory of Special Relativity

RIVERS OF TIME THE THEORY OF SPECIAL RELATIVITY Throughout the 19th century, a lot of attention had been given to the propagation of electromagnetic radiation (light). Most scientists believed that space comprised an aether - a medium that provided the characteristics needed to carry electromagnetic waves. A lot of experiments had been conducted to detect this medium; but, the aether was never found. The progression of work by Lorentz, Poincare' and many others, which grew out of the aether pursuit, led to the discovery of relativistic phenomena. Einstein, in his famous 1905 paper, postulated the Principal of Relativity as presented by Galileo, but also incorporated the constancy of the speed of light. The principal of the fixed speed of light required a complete abandonment of the aether theory. Einstein provided, through elegant simplicity, a theory that supported the findings of many great scientists of that century. The propagation of electromagnetic waves has a unique characteristic... Every observer will measure the same speed for light regardless of his or her motion relative to the the source. The speed of light c in a vacuum is 3x108 meters/sec, or about 186,000 miles/sec. There are many consequences to the constancy of the speed of light, and this was the idea surrounding Einstein's theory. I will present a slightly different prospective, and greatly simplified variant, on the Theory of Special Relativity. It applies the concept of temporal space and is based on Minkowski spacetime. Hermann Minkowski was a German mathematician who -shortly after the advent of the Relativity Theory- realized that, through mathematical manipulation, time could be treated as another dimension in space. Extending our familiar 3-dimensional space to incorporate a 4th temporal dimension meant that simple orthogonal transformations (a means of moving an observational prospective from one Cartesian coordinate system to another) could be applied to problems in Special Relativity. It was the interleaving of space and time, created through such transformations, that gave rise to the mathematical concept of a spacetime manifold. Experiencing time is like being afloat a fast moving river. The flow is continuous, and never-reversing. Movement through time is somewhat analogous to movement through space; but, the difference being that we are progressing through a type of space that exists beyond human perception. It is the eternal motion along this temporal river that gives us a sense of time. The odometer used to track our progress is, of course, the clock. If motion along the temporal pathway is slowed, time itself would decelerate. Clocks would begin to lose time, the rate of nuclear decay would diminish, and the biological aging process would slow. Time dilation (the slowing of time) is one of the consequences of the Special Relativity theory. Its effects have been demonstrated in laboratories around the world. This is not the stuff of science fiction! The Theory of Special Relativity is generally applied to physical systems that move with uniform motion. The conceptual platforms where observers and clocks reside are referred to as inertial frames. Because uniform motion implies movement along a straight-line path; we can describe uniform motion in spacetime using two mutually perpendicular velocity vectors; one defining motion in ordinary space and the other defining motion through time (temporal space). These velocity vectors can then be combined to produce a singular resultant vector, which I'll call the spacetime velocity vector. [See figure 1] As a basic premise of this Special Relativity presentation, the magnitude of the spacetime velocity vector will be held constant and equal to the speed of light c. Otherwise, the two constituent velocity vectors (in time and space) are interdependent. By increasing the magnitude of one, you cause a decrease in the magnitude of the other. An object at rest will simply move through temporal space at exactly the speed of light. As the object begins to move through ordinary space; its speed through temporal space begins to decrease. When Albert's spaceship left the earth and traveled towards the nearest star; the motion of his spaceship, relative to earth, cause time to progress more slowly for him and any passengers who were on board... again, relative to the passing of time on the earth. This is the effects of time dilation. We can take this idea one step further. Albert and his spaceship passengers would find themselves arriving at the destination in less time than was originally planned. This is a logical consequence of time passing more slowly on the spaceship. The passengers do not feel the effects of time dilation; so, their early arrival might be interpreted one of two way: (1) the spaceship must have been moving faster than was expected, or (2) the distance traveled to the destination was shorter than was expected. An earthbound observer measures the spaceship's speed vx, as it travels out into space. Likewise, a passenger on board the spaceship watches as the earth recedes from behind, and measures its speed to be vx. (This is the reciprocity of relative motion.) Thus, the first interpretation could not be correct. From the prospective of the space-traveling passengers; space must then be compressed along the direction of travel. This is the effect of spatial contraction. It is, yet, another consequence of the Special Relativity theory. If the speed of an object through temporal space is plotted as a function of its speed through ordinary space, keeping the magnitude of the spacetime velocity vector fixed; the graph would map-out one quadrant of a circle. Any object's resultant motion through spacetime would, necessarily, fall upon this curve. If the speeds of two objects through ordinary space were vx1 and vx2, and the corresponding speeds through temporal space were vy1 and vy2, respectively and, for example, vx1 < vx2; then vy1 > vy2. [See figure 2] There may be a point of confusion regarding velocities measured in temporal space. Velocity, by definition, is the distance covered in a given time and in a given direction. It may seem senseless to apply velocity when describing motion through time. Hopefully, some confusion will be eliminated by remembering that velocity and distance are quantities that are usually measured by an observer who is at rest. In the case of temporal velocity measurements; the rest observer's clock will be applied in the calculation. We can choose to utilize the clock reading, directly, in one inertial frame, but then utilize the concept of temporal distance to gauge time in another inertial frame. Special relativity does not assume that the passage of time is the same across all inertial frames. Thus, we can treat time differently between the two systems. The highlighted area below provides a mathematical treatment of time dilation and spatial contraction. This section can be skipped without loss of comprehension. Following this analysis, I'll revisit our traveling emissary, Albert, and describe his experiences with time dilation and spatial contraction. Time is just a measure of distance in temporal space. The time interval ∆T, displayed on a clock, is directly proportional to the distance ∆y traveled through temporal space. κ is simply a proportionality constant. ∆T = κ ∆y   (Eq 1) Let vx represent the speed of an object moving through ordinary space and let vy represent the speed it progresses through temporal space. The object’s resultant speed (the magnitude of the spacetime velocity vector) must always be equal the speed of light c. (vy)2 + (vx)2 = c2   (Eq 2) In the special circumstance where the object is completely at rest in ordinary space; its speed through temporal space can only be equal to c. vy = c, when vx = 0 Let ∆To represent a time interval measured by an observer who is at rest. ∆yo is the distance this rest observer progresses along the temporal dimension during the time interval ∆To. A clock held by a rest observer is said to keep proper time. ∆yo = c ∆To   (Eq 3) Using Eq 1 and Eq 3, the proportionality constant κ is found to be 1/c. Eq 4 is just a rewrite of Eq 1. 1/c = ∆To/∆yo = ∆T/∆y = κ ∆T = (1/c) ∆y   (Eq 4) Because the magnitude of the spacetime velocity vector is restricted to the speed of light, the temporal speed vy can be written as a function of the spatial speed vx. vy = [c2 – (vx)2]½   (Eq 5) Through the inspection of Eq 5, we can predict that greater speeds in vx will result in lesser speeds in vy. As the speed of an object within ordinary space approaches the speed of light; that object’s progression through temporal space approaches zero. When movement through temporal space is halted, time itself would cease to exist. If the moving object were a person, he or she would stop aging. Clocks would stop running, and all motion and activity at a subatomic level would cease. The object or person would be in a state of suspended animation. It should be noted that the clock carried by the moving object would appear frozen only from the prospective of the rest observer. Proper time is measured by the observer’s own clock, and that would be unaffected. The moving inertial frame carries its own time along with it. A clock sitting inside a moving inertial frame measures time interval ∆T, whereas a clock held by the rest observer measures a proper time interval ∆To. Clearly, the time intervals are not, necessarily, equal! Time dilation is determined by the ratio of (∆T/∆To), which is a function of the spatial speed vx. Based on Eq 4, it is determined that time dilation can also be represented as a ratio of temporal distance (∆y/∆yo). This is why time and temporal distance are interchangeable. (∆T/∆To) = [(1/c) ∆y]/[(1/c) ∆yo] = (∆y/∆yo) Time intervals ∆T and temporal distance intervals ∆y are all measured relative to the rest observer. Thus, the relative measurements are calculated as ratios. Temporal distance intervals are functions of spatial speed and proper time. ∆y = vy ∆To = [c2 – (vx)2]½ ∆To ∆yo = c ∆To (∆y/∆yo) = [c2 – (vx)2]½ / c = [1 - (vx)2/c2]½ (∆T/∆To) = (∆y/∆yo) = [1 - (vx)2/c2]½   (Eq 6) Eq 6 specifies the time dilation effect. If the object's speed, relative to the rest observer, approached the speed of light; the time interval ratio (∆T/∆To) would approach zero. From the prospective of the traveler, the apparent spatial distance ∆xm traversed to the nearest star would be smaller than the distance ∆xo measured to the star by an earth-bound observer. Eq 7 represents spatial contraction along the direction of travel. ∆xm = vx ∆T = vx [1 - (vx)2/c2]½ ∆To = [1 - (vx)2/c2]½ ∆xo   (Eq 7) In summary, the time dilation and spatial contraction formulas are repeated below: Time Dilation:   ∆T = [1 - (vx)2/c2]½ ∆To Spatial Contraction:   ∆xm = [1 - (vx)2/c2]½ ∆xo In part 1, we learned of a space-traveling emissary named Albert who was making a trip from his home planet Earth to the nearest star Alpha Centauri, which is located 4.37 light years away. A light year is defined as the distance light propagates in one-year, (1 light year) = (c)(1 year). If we assume that Albert travels to his destination at the constant speed of 90% the speed of light; everyone would expect his trip to take (c)(4.37 LY) / (.90 c) = 4.86 years. However, Albert's experience will be very different due to the effects of time dilation and spatial contraction. In this example, Albert is moving toward Alpha Centauri, relative to the earth, at a speed that is close to the speed of light. The passage of time would occur more slowly for Albert, relative to the passage of time on Earth, due to this high rate of motion relative to the Earth: ∆T = [1 - (vx)2/c2]½ ∆To = [1 - (.90 c)2/(c)2]½ 4.86 years ∆T = ([1 - .81]½)(4.86 years) = (.19½)(4.86 years) ∆T = (.436)(4.86 years) = 2.12 years Albert would only clock 2.12 years during his trip to Alpha Centauri, whereas an observer on Earth would have clocked 4.86 years. If an observer on Earth could somehow peek into the spaceship and view Albert's activities along the way; the observer would see the world inside the spaceship moving in slow motion. Clocks on board would tick at about 43% the rate of clocks on earth and Albert's biological systems would slow by the same rate. From Albert's prospective, time runs normally. He would realize that the spaceship did not travel the expected distance of 4.37 light years. Space was considerably compressed along the direction of travel, as experienced by Albert, which reduced the distance traveled to the star: ∆xm = [1 - (vx)2/c2]½ ∆xo = (.436)(4.37 LY) = 1.91 LY Albert would only need to travel 1.91 light years to get to Alpha Centauri, and he would do it in only 2.12 years! There is an apparent paradox in Albert's situation, however. He could have considered himself to be at rest and the Earth racing away from his spaceship at 90% the speed of light. In this situation, he would expect the clocks on Earth to run at about 43% the rate of clocks on his spaceship (the opposite effect). Both the Earth-based observer and Albert are absolutely correct in their assessments of time dilation, even when the results seem paradoxical. This is the bizarre nature of our universe! There are two separate realities in play, and there would really only be a paradox if and when we attempted to land Albert back onto the Earth. That, however, would require turning him around and violating the rules of uniform motion. The interpretation of dual realities is a perplexing phenomenon and will be investigated further in part 4 (Parallel Universes). Einstein realized the limitations of the Special Relativity theory and needed a more generalized theory to account for non-uniform motion. The Theory of General Relativity took into account gravitational forces and the forces caused by non-uniform motion. In the next installment, time dilation and spatial contraction will be calculated for our space-traveling emissary Albert when his spaceship accelerates and decelerates during his journey. We'll also discover that the effect of time dilation is evident, even when Albert returns home to planet Earth!