## Special Relativity

Special relativity accounts for differences in time measurements made by an observer travelling with a moving spacecraft and measurements of time made by a second observer at rest. Because all velocities are assumed to be relative, there is the option of deciding that the second observer is moving and the spacecraft is stationary. Spacetime is assumed to be infinitely divisible and there is no concept of absolute rest.

Quantized spacetime offers an approach based on a spacetime that is made up of spherical spacetime quanta with the smallest possible dimensions in space and time. Tightly packed into a four-dimensional lattice, these quanta are anchored to the local gravitational field. This in turn is set in the overall gravitational field of the universe. Clearly, there are dynamic changes in the field as different bodies move around their orbits, but these fluctuations are readily accounted for in precise navigation systems. Consequently, the lattice can be considered a dynamically stable reference grid over distances of interest. And a particle that is not moving from one quantum to another can be considered at rest.

The ability to distinguish between objects at rest and objects in motion reveals the presence of two different type of time in the lattice. A particle or configuration of particles at rest in the lattice experiences the intrinsic time steps that occur across the lattice as a whole. This is the intrinsic time in which we describe the history of the universe. Objects at rest age at the rate of intrinsic time, which amounts to passage of one Planck time with each intrinsic time step. When a particle moves from one quantum to the next, it requires a time step to make the transition, involving activity in two quanta. This action in transfer requires time that would otherwise constitute the intrinsic time experienced by the particle.

Transfer time steps have the same length as intrinsic time steps, to maintain synchronization in the lattice. Their interpolation among intrinsic time steps when a particle moves lengthens the passage of time between each intrinsic time step. The effect is to slow down intrinsic processes such as particle disintegration or completion of chemical reactions between particles. A muon with no internal structure, for example, disintegrates after about 2.2 microseconds when it is at rest. If it is moving at 99.3 per cent of the velocity of light, this lifetime is extended nearly nine times longer. At the same speed, composite body, such as a neutron made up of quarks and gluons, or a molecule made up of atomic nuclei and orbiting electrons, would also find decay or other intrinsic chemical processes similarly extended in time.

Recall that the transfer of a particle to the next quantum requires movement of one Planck length in one Planck time. This corresponds to the speed of light, so that the speeds we are familiar with, far removed from the speed of light, are achieved by a particle spending many time steps stationary in intrinsic time. Even as a particle’s average speed approaches the speed of light, many quanta remain available for interpolation of transfer time steps.

The photon, the particle that always travels the speed of light, is an interesting anomaly in this picture of intrinsic and transfer time steps. Its intrinsic process is not decay or transformation into another particle but ceaseless transition from one quantum to the next. In every intrinsic time step of the lattice as a whole, it transfers from one quantum to the next. This intrinsic-transfer time duality is appropriate for a particle that is its own antiparticle.

Given that intrinsic processes like radioactive decay slow in the presence of motion, one might expect to demonstrate this by taking such a material up to the space station orbiting at 7.7 km/s and measuring the slowing by means of a clock. Unfortunately, the clock would slow at exactly the same rate and no change in the rate of decay would be apparent. This reveals a fundamental feature of motion at a constant velocity, it cannot be detected by internal means alone.

### Slowing of Time

In fact, the slowing of time in special relativity theory is readily developed directly from the effect of uniform motion on a clock. A clock is chosen whose internal workings are clearly apparent. As shown in Figure 1, it consists of a light source that projects a beam vertically from a base to a fixed mirror, that reflects the beam back to a second mirror, at the source, which reflects the beam up to the top mirror, the process repeating cyclically.

*Figure 1 A light clock. Mirrors M1 and M2 are firmly attached to a
framework which maintains a fixed vertical separation, L, between
them. Light from a source at M1 illuminates M2 and returns to M1. *

Passage of the light from source mirror, M1, and back from the upper mirror, M2, represents two ticks of the clock. If light from M1 travels a distance L up to M1, the time period for one pass of the light between the two mirrors is T, where

T = L/c seconds.

If the distance between the mirrors is one meter, the period between ticks is just over 3.3 nanoseconds.

To measuring the effect of motion on time, a stationary observer has one light clock, the other is mounted in a spacecraft. The two clocks are identical and register the same time interval when the spacecraft is stationary. Figure 2 shows the light path in the spacecraft clock as viewed by a stationary observer when the spacecraft is moving at v m/s. This observer sees that the light in the clock must travel along line ad, if it is to be reflected by the upper mirror as it moves along with the spacecraft. The length of this line, D, is clearly longer than L. As light travels at a constant speed, c, the stationary observer concludes that light should take longer to travel between the two mirrors.

But the spacecraft crew is moving along at the same speed as the clock, so to them the light still appears to be traveling the same vertical distance L, apparently ticking at the same rate as when the spacecraft was stationary, and apparently registering the same period of time with each tick.

*Figure 2 The light clock on a spacecraft moving at velocity v. By the
time light from M1 reaches M2, M2 and the spacecraft have moved forward a
distance vT’. T’ is the new time period for light to travel from a to d.*

To reconcile the views of the crew and the outside observer, both must recognize the spacecraft clock is ticking at a slower rate, with a time period between ticks of T’. The increase in the duration of each tick is such that for the same number of ticks, the total time becomes sufficient for light to travel the new distance D at c. That is,

T’ = D/c

In the time light travels the distance D, the spacecraft moving at velocity v is observed to move a distance x to the right (line ab). It is a fundamental assumption in special relativity that this motion cannot be detected within the spacecraft without recourse to external clues. However, the external observer readily perceives

x = v T’

The mirror remains at a distance L from the source (line sq). By Pythagorus, the length D is given by

D2 = L2 + x2

Or D = sqrt( L2 + (vT’)2 )

If D is entered into the equation for T’ the new time interval is found to be

T’ = T/(1-v2/c2)1/2

This is the expression in special relativity for the intrinsic time dilation experienced at velocity v.

### Time Dilation in QST

In the QST approach, if there are N quanta per meter, then the length L of the light requires LN quanta. These are the intrinsic quanta required by the photons, and unless the light path shrinks, their number does not change.

When the spacecraft has a velocity v, the length of the light path increases to D, which requires DN quanta.

With this velocity, the spacecraft is observed to move a distance x to the right (line ab). This requires Nv transfer quanta.

The distances specified by the quanta continue to be related as a Pythagorean triad, so that

(ND)2 = (NL)2 + (NvT’)2

Or D2 = L2 + (vT’)2

If D is entered into the equation for Δt’ the new time interval is found to be

T’ = T/(1-v2/c2)1/2

For the crew, the effect of the stretching of the light path to D is not immediately obvious. It is achieved by interpolating transfer quanta between intrinsic quanta, spacing them further apart in time, throughout the moving assembly. The interpolation causes a longer time to elapse before each intrinsic process in the spacecraft and crew completes. Yet, because every element of the spacecraft and crew experiences this time dilation, everything appears to be continuing at the normal rate. Neverthless, biological processes of aging are intrinsic processes. So, the crew’s rate of aging at a given velocity slows compared with that at rest. Astronauts on the International Space Station age about 7 milliseconds less than their counterparts on Earth over a period of 6 months, due to this relativistic time dilation.

From the equations, it is clear that the time dilation due to relative motion as forecast by special relativity for continuous spacetime will remain the same for absolute motion as forecast in quantized spacetime, as the spacetime quantum is an alternative space measurement unit directly related to the meter. In a similar way, the other equations of special relativity will be reproduced by quantized spacetime.

It is possible that the principle that absolute velocity is not detectable within the spacecraft may eventually be invalidated by advanced instrumentation able to navigate by following a map of the local gravitational field. This will penetrate spacecraft as there is no shielding against gravity. Detection at the required sensitivity is impossible at present due to the very low strength of the gravitational field (or the low mass of the proton). But the possibility that a new method of detection might be found in the distant future cannot be ruled out.