Special Relativity
Special Relativity
Special relativity recognizes that observers moving at different uniform velocities will find that time and length measurements made in the other's environment will differ from their own view of those measurements. Equations based on the velocity difference and the speed of light enable translation of space and time dimensions between the two environments [13]. Such environments where gravity and acceleration are absent are called inertial reference frames. The theory is based on the constancy of the speed of light and holds that an experimental setup in a moving inertial reference frame finds that the same physical laws apply as when it is not moving. Rephrasing, one cannot discover one's own velocity by internal measurements alone. That velocity has to be measured relative to some external reference, which may also have a velocity.
This is a challenge for a description of spacetime based on a uniform lattice. One might think that such a lattice would provide a reliable frame of reference within which everyone could agree on times, locations, and transfers of measurements between them. However, this is not the case. There is no way of detecting a fixed point in the lattice to use as a reference. The lattice only reveals itself by its interaction with matter and radiation. So the only reference for velocity is matter or radiation that is itself moving through the lattice. Which takes us back to the finding that velocity measurements have to be made relative to some external marker (which means of course, some other object in the lattice, not the lattice itself).
The way the lattice handles the phenomena of special relativity can be portrayed by the example of an astronaut piloting a spacecraft at high speed past an observer who regards himself at rest. The observer and the astronaut both have accurate clocks. To investigate the relationships between velocity and both time and space, the clocks are in the form of a light beam passing vertically back and forth between two mirrors. On the spacecraft, the mirrors are on the floor and ceiling. On the ground, the clock stands next to the observer; the mirrors are held apart the same vertical distance as on the spacecraft. When the spacecraft is stationary in front of the observer, light takes the same time on both clocks to go from a source at the bottom mirror up to the top mirror and back down to the bottom mirror.
The ticks on these clocks are the time steps of the spacetime quanta traversed by the light beam. One tick corresponds to the Planck second. When the observer and astronaut are both stationary, they count the same number of ticks for one travel up and down by he light beam. Let us call that the bounce time.
The spacecraft takes off and flies a previously measured distance across the observer's field of view at a constant speed. Both he and the astronaut record the bounce time for the traveled distance on their own clocks. They each find their own clocks are operating normally. Their individual pulse rates, for example, are within the usual ranges.
The stationary observer, however, thinks something important has changed in the spacecraft clock because of the spacecraft speed. He sees that as the spacecraft moves forward, the light rising from the source by the floor mirror has some extra distance to travel to catch up with the mirror on the ceiling. He realizes that the light has to travel at a forward angle to the vertical to intercept the upper mirror: by the time the light hits the mirror, the light has reached a new position in space. And he sees that light has to follow a similar forward-leaning angular path to reach the new position the lower mirror arrives at after moving through space. He concludes that spacecraft motion causes light in its clock to travel an inverted v-shaped route that is longer than the purely vertical route in his own clock.
As light travels at a constant speed, the ground observer assumes that the longer path causes the spacecraft clock to measure a longer bounce time. From the spacecraft speed, which he has measured, he calculates the new path length together with the number of extra ticks it will cause in the bounce time of the spacecraft clock. He relays his finding to the astronaut. No way, she says. Her clock is still producing the same number of ticks per bounce as when she was on the ground.
The astronaut's experience is exactly what special relativity requires. If she were able to detect a change in the bounce of time of her clock at her new speed, the clock could function as a speedometer: she could measure her speed by internal means alone. Special relativity holds this to be impossible.
The ground observer tries to figure out how to reconcile his view of the time he thinks the astronaut's clock should be showing with what the astronaut is actually seeing. He concludes that the spacecraft has shrunk in the direction of motion, so that the inverted v-shaped path has become the same length as the path in his clock. Then light will take the same time to travel this path, and the number of ticks registered by the two clocks will be the same. It means that for him, the spacecraft clock is running slow compared with his clock. He figures out the necessary contraction with a calculation that involves the speed of light and the velocity of the other clock.
The astronaut congratulates him on his solution. She points out, however, that her clock has registered the same number of ticks. This indicates the number of spacetime quanta she has passed through has remained the same. If the spacecraft has contracted, then each spacetime quantum will have had to become shorter. So, if the ground observer believes the speed of light is constant he will have to shorten the time interval of the spacecraft quanta to match the new quantum length. The new length divided by the new time interval must equal the speed of light.
The calculation of changes in length and time with velocity relies on simple Pythagorean geometry. The conversion equations make use of the speed of the spacecraft and of light; the equations are well known, and have proved accurate in innumerable tests. The equations of special relativity show that if the spacecraft were to reach the speed of light, the time interval would become zero and time would stop. Similarly, the space interval would become zero and space in the direction of motion would disappear. However, the equations of relativity also show that the effective mass of the spacecraft increases with relative speed, so that it too would become close to infinity as the speed of light was approached. There is never enough energy to accelerate the growing mass to light speed.
The astronaut is sure her clock kept good time and no shrinking of the spacecraft, or herself, took place in the direction of flight. The number of quanta has not changed. When she returns and notices the ground observer has apparently aged slightly more than she has, she puts this down to the strain of trying to understand his own observations.
In terms of the spacetime lattice, the relative motion decreases the apparent size of the Planck time and the Planck length, while maintaining a constant velocity of light. The changes external observers attribute to the spacetime lattice are necessary to interpret phenomena in the moving spacecraft or in any other moving inertial reference frame. But these do not constitute a real change in the lattice at the spacecraft. There could be ten different observers with ten different relative speeds to the spacecraft, each with a different view of the changes in spacecraft time or length, not to mention the even greater multiplicity of views when the observers observe each other. Translations between all these views by all of the astronauts are made accurately with the same equations of special relativity. To any two observers the effects are real. It's just that reality is relative as far as velocity is concerned.
From these principles of special relativity, the effects of gravity on spacetime can be examined in the next section.
1/12/2016