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.
