Relatively Simple

Relativistic time and space dilation made easy

Relativistic time and space dilation seems like a difficult and complicated topic to comprehend.  It’s not.  It’s actually quite simple.  I have no idea why scientists and textbooks make it seem difficult.  The following is my simplified explanation, requiring nothing beyond a knowledge of high school algebra and geometry.

Relativity – the study of space-time in relation to speed and gravity.

Space and time are really the same thing.  The exact same thing.  Accept it, because it’s true.  They are inseparable parts of the whole of reality.  To cross a certain amount of space, you need a certain amount of time.  Given a certain amount of time, you can cross a certain amount of space.  It is somewhat mistaken to even call them separate things, so they are referred to collectively as space-time.

Nothing moves faster than light in a vacuum because it moves at the fastest speed possible – the rate of one unit of space per one unit of time.  Light can do this because it has no mass, and is a special case.

The perception of space time shifts as a particle changes speeds.  (I speak of speed rather than velocity because, for general purposes, the direction it is traveling does not matter – just the scalar of it’s speed.)  The faster the particle travels, the smaller space appears to it, the the slower time seems to pass for it.  This, as a particle speeds up, space-time appears to shrink.  This has a limit at the speed of light – where time appears to stop, and space appears to shrink down to nothingness.  After all, if there is no time passing, no distance can be traversed.  Please keep in mind that this is the perspective of the particle itself – in reality, it is moving amazingly quickly across vast expanses of space.

The formulas used to describe the dilation of space-time by speed are based on the Lorenz factor:

y=1/sqrt(1 - (v^2 / c^2))

Where v is velocity (speed) and c is the speed of light.  This can be simplified greatly if we choose to use the ‘natural units’ where c = 1, and measure v in terms of c.  This results in:

y=1/sqrt(1 - v^2)

Of course, what this results in is a multiplier – how much slower time seems to pass, how much shorter space seems to get.  This can be simplified greatly by inverting it:  what fraction of ‘normal’ space time is the perceived space-time?  This inversion results in the following:

t=sqrt(1 - v^2)

Where 't’ is the perceived length of space-time as a fraction of ‘true’ space-time, and v is the speed as a fraction of the speed of light.  If this looks familiar, it’s because it is.  It’s the solution to how to graph a circle.

y=sqrt(1 - x^2)

Which, of course, derives from a much more recognizable formula (assuming a center at origin (0,0) for simplicity):

x^2 + y^2 = r^2

This is also the Pythagorean Theorem solution to all right triangles:

a^2 + b^2 = c^2

Given that what we are dealing with are speed, perceived space-time, and the speed of light, this works out to be:

v^2 + t^2 = c^2

Since we’re using the natural units where c = 1, this simplifies even further to:

v^2 + t^2 = 1

It’s easier to see this as a graph.

In this framework, the maximum possible length of any side is 1, and the hypotenuse, representing c, is defined as having a length of 1.  It is then quite easy to see that as speed increases, perceived space-time must decrease.  The relationship between these two variables can also be plotted out quite easily, as the first quadrant of a circle.

So, as can easily be seen, a particle must be moving quite quickly before the space-time distortion becomes really noticeable.  Also note that the radius of the arc is defined by c, the speed of light.  At rest, v=0, a particle experiences t=1, full space-time.  At the speed of light, v=c, a particle experiences t=0, collapsed space-time.

Gravity distorts actual space-time, just as velocity distorts perceived space-time, and in exactly the same ways.  This results in these two graphs, where g is the pull of gravity, expressed as velocity.  (Justification – gravity is normally represented as acceleration, d/t², which is velocity applied over time.  For this discussion, I am interpreting the pull of gravity as an instantaneous velocity, d/t, with no passage of time to aggregate its effects into acceleration.)

This shows g distorting space-time.  Now we get to the fun part.

For the following, I make one assumption:  space-time is itself subject to quantum theory.  In other words, space-time is digital, not analog.  There are discrete points of space-time, separate and somewhat independent from each other.

‘t’ can be most easily understood as representing the distance between points of space-time.  No, there isn’t anything between these points.  The distance between points is a measure of the length of space-time – this is physical length, as well as the length of time.  It takes time to traverse space, and space for time (change) to happen in.  Time and space are the same thing, remember.

Scientists have already proposed a minimum possible length – the Planck length.  Quantum effects dominate at this scale.  I propose that each point of space-time is discrete, and has its own properties.  For our purposes here, I am limiting the discussion to gravitic effects.

This interpretation shows that the curvature of space-time is really a simple change in density.  The higher the local gravity field, the closer the points of space-time are to each other.

Taking a look at the graph of gravity, notice that it arbitrarily stops at g=c=1.  This is the event horizon of a black hole.  However, there is no reason for gravity to arbitrarily stop there.  Gravity keeps increasing in black holes that are larger than the bare minimum.  So, inside the event horizon, do the equations and graphs break down?  Not at all, if you simply convert your units of measurement to the set of complex numbers.  Normal space is restricted to the subset of complex numbers better known as the real numbers.  Inside the black hole, imaginary numbers rule.

Notice, then, that the higher the gravity, the larger the imaginary component becomes.  This implies that, given sufficient gravity (g > sqrt(2)), the black hole becomes larger on the inside (on the imaginary plane) than it is on the outside (on the real plane). 

But what of velocity?  On the real plane, v is always less than or equal to c.  However, inside the black hole, space time has an added dimension provided by the complex plane.  So there, and only there, velocity can acquire a complex component.  The complex component could, theoretically, become arbitrarily large.  v^2 + t^2 = c^2 (where c=1) still holds true, because the squared imaginary plane components become negatives, and subtract themselves back out, leaving a result of 1.  The same also applies to the gravity effects formula of g^2 + t^2 = c^2.

Other musings.

These are things I cannot yet prove, but the logic is there to strongly suggest, and derive from valid observations.

There are three basic layers to reality.  The physical layer (quantum particles, having their own properties such as velocity and charge), space-time (quantum points, having their own properties such as distance apart), and fields (analog equations).

Fields are not a part of space-time, and are not subject to its restrictions.  Fields propagate arbitrarily (infinitely) quickly over arbitrary (infinite) distances.  Examine – given the theoretically infinite space-time density at the event horizon of a black hole, where g-c and t=0, if fields such as gravity and electro-magnetism were subject to space-time restrictions they could not propagate any further.  Yet, we know that black holes have gravitic and electro-magnetic effects outside their event horizon – that is, in fact, what makes them interesting.  And it has been proven that the earth is attracted to where the sun actually is at any given moment, not where it appears to be in the sky due to the over eight light-minute distance.  Therefore, the fields themselves must not be subject to normal space-time restrictions.  Occam’s razor dictates that we take the simple solution that fields propagate infinitely quickly.   

Points of space-time are discrete, and have properties.  They grow closer to each other in the presence of a high gravity field.  Thus, we easily see that the points can move relative to each other.  The next logical step is to interpret the points of space–time as a fluid, instead of a fixed grid.  They can move around, and even past, each other.   They can move in different directions.  Their movement is dictated solely by changes in the local gravity field.  This helps explain why the outer reaches of spiral galaxies spin at the same rate as the centers – the points of space time have been caught up in the spin, as well as the actual particles of matter themselves.

Points also exhibit the properties dictated by the local fields.  The gravitic field at a given point dictates that a point has a vector property corresponding to its strength and direction.  In other words, each point of space time imparts a motion vector to any particle of matter that happens to reside there.  (I suspect that the same applies to electro-magnetic effects, but that topic is beyond my understanding.  But, as a simple matter of symmetry and common effects, it should apply.)  The final motion of a particle is a combination of all the vectors acting upon its own momentum vector.

The behavior and properties of matter define their effects on fields.  This is a known, basic, given fact.  Fields affect the points of space-time.  Space-time properties affect the particles of matter located thereon.  And so on, and so on, in a perpetual game of rock-paper-scissors.

Edit 2020:  Looking back, I'm almost there.  Space is not discrete, but particles sort of are.  There is no gravitic field, because gravity is not a force.