A simple note on SpaceTime. Note 1.

Date : 16/12/2017
Version: 0.2
By: Appie vd Sel
Status: not ready yet
Remark: Please refresh the page to see any updates.




Fig 1: Just an illustration of some views on SpaceTime.




It's nice to talk about some interesting fields in Physics.
However, I have no doubt that you will qualify this text as fairly "high-level",
and not much "in depth".

Quite a few new theoretical insights have emerged, say, as from the early '80's (or so), on the "structure"
of SpaceTime. Why I say SpaceTime, instead of just Space, is indeed somewhat debatable.
I try to illustrate that in a moment from now. It is true that the ideas of Einstein, Lorentz, Minkowski
and others, in the first decades of the 1900's, showed that Time and Space are indivisible parts
of the same "something", called SpaceTime.

Later, physicists always have dreamed to reconcile "General Relativity" of Einstein, with Quantum Mechanics.
Both are pretty old theories by now (developed in the first halve of the former century), but they still form
the very fundaments of physics.

For example, most people view Space as a true continuum. As an example that this is completely reasonable,
is that on a human scale, Space is "smooth" and there seems to be no reason to say that it has some
discrete, or sort of granular, or sort of lattice form.

However, if you consider Quantum Mechanics, which works very well in the (sub-)Atomic domain, and with
waves and particles, then it turns out that discrete states, and quanta, and eigenvalues,
and specific quantum numbers, seem to dominate the theory.
What's more, the theory is in accordance with observations and experiments.
So, it works very well with events, objects and observables, on a very small scale, with matter and radiation.

From such a perspective, it's not so strange that physicists, a long time ago, might thus have considered the idea,
that even quantization of "Space" might be true too. As if Space has a sort of "lattice" structure.
This then might be a reality on an extremely small scale, so that on larger scales (if you zoom out),
Space just simply "looks" like a classical continuum again.

Early ideas of Penrose, and a modern theory such as "Loop Quantum Gravity", indeed have provided
a physical framework (a fairly complete theory) for such ideas.

But physics is a strange adventure. Indeed, there exists quite a few alternatives (or slightly different
alternatives) for something like "Loop Quantum Gravity", like other unifying "Quantum Gravity" theories.
One other rather popular alternative unifying theory, is "String theory".

But some newer highly remarkable (almost exotic) ideas did hit the community of physicist too, during the 2000's.
Today, there exists quite a few "streams", within (most notably), theoretical physicists.
Some have published articles, saying that Space, when considered "in depth", is actually build from extremely
small "wormholes". Some very well-kown theoretical physicist, support such ideas.

Others, and this seems to get more and more consensus as time goes on, has made it plausible
that a feature called "entanglement", rules (or determine) Space or SpaceTime (and gravity).

And if the above was not enough, something else is going on too.
As of the '80's (or so), some hints from Quantum Gravity / Lattice theories, and even nummerical theoretical
calculations (based on very specific theorems), seem to indicate that the number of dimensions (d)
might depend on "scale" (dimensional reduction). In the limit, when the scale of SpaceTime goes to the minimum,
d goes to "2". In Jip and Janneke language: as if a surface is the most basic unit.

Needless to say it all still is very hypothetical, but those hints are not mere speculations (!)
You might wonder, as I do, how such statement as listed above, relate to the "curled up"
extra dimensions of String / M theories.
Anyway, I like to describe (in a very lightweight way) such hints as mentioned above.
Indeed, Physics today, is truly very exciting. But the "big picture", still is quite incomplete.

It's also nice to say some words on phrases as "canonical" and "covariant" formulations of a theory, and the meaning
of "background" (or: How to get the SpaceTime out of the Theory)

It's pure fun to take a closer look at such theories, and this is what I will try to do in this note.
This note will be then be further characterized by a fairly large number of short chapters.

However, this will be an insanely simple note, so please keep that in mind....
And, I think "in format" it will be rather ugly, since I never have a pre-determined "plan" for a note.

1. SpaceTime in Relativity.

The considerations in this section are more conservative in character.

However, we will also touch on a subject like "Lorentz violation" which is a very interesting field indeed.

Einstein produced two magnificent theories in the early 1900's: The "Theory of Special Relativity" (1905),
and the "Theory of General Relativity" (1916), both often abbrieviated with "STR" and "GTR".

Both theories are highly involved in discovering the properties of SpaceTime.
Ofcourse, both theories are absolutely monumental! I can only distill a few points from those theories,
which is what I am going to do here.

1.1 The common (Regular) 3D coordinate system:

Ofcourse we can visualize a three Dimensional Cartesian coordinate system, using an x-axis, y-axis, and z-axis, all
perpendicular to each other. Nothing special here. This is highschool math. In that "3D space",
points can be described by (x,y,z), where x, y, and z can take on any value.
The x, y, and z, are "spatial" meaning that they are also involved in something called a "metric",
which you often can relate to the fact that you are able to define a distance between points.

For example, between the points (x1,y1,z1) and (x2,y2,z2),
you can draw a linesegment, which also means that we can speak of the distance "d" between those two points.

Simply using the Pythagorean theorem, the distance "d" squared, is:

ds2 = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2=
dx2+dy2+dz2.

By the way, in math nothing prevents you from using e.g. a 6 dimensional space, where points might be described
as a 6 tuple (in general an n-tuple) like (x1,y1,z1,u1,v1,w1).

1.2 4D SpaceTime (Minkowski SpaceTime):

3D space and time, together form a 4D SpaceTime. But how to really define it in terms of, say, "points"
in such a space, just like we did above? First, I must say that 4D SpaceTime, is not just like adding
one extra spatial dimension to 3D Space (like if you would go from 2D space (x,y), simply by adding the z-axis,
in order to arrive to 3D Space).
No, it must have a time "t" related dimension. But, If we would simply use (z,y,z,t), then we would not be able
to get a "metric" as we saw above (like a distance "d" between points).

Now, in order to let the fourth coordinate, relate to a spatial dimension, we can use "ct", where "c" is the
universally constant speed of light. A simple illustration might help: You know that if you bike 10 m/s,
then after 5 seconds, you have covered 5 x 10 = 50m. So with a constant speed, distance = speed x time.
In order to correctly indentify points (actually "events", about later more on that), we thus might use:

(x,y,z,ct) or, which is not much different, (ct,x,y,z), which you may also find in the literature

In such a case, we are able to use a metric (like a "distance") between points in such SpaceTime:

ds2 = -c2t2 + dx2 + dy2 + dz2

Note: there is a little issue with the story above. In the true metric "ds" (distance), the c2t2
term must indeed be negative. I left out to correctly explain that.
However, my simple description of Minkowski SpaceTime, will already help us in a discussion of "events".

1.3 A few highlights of STR:

STR is mainly involved in "frames of reference" (coordinate systems) which move with a uniform or constant speed
with respect to each other.

Guided by reasonable assumptions, Einstein reasoned that:

-The laws of Physics should hold everywhere. The law of physics are the same in any frame of reference.
-There is no preferred "direction" in true space. Or Space itself is homogeneous and isotropic.

-And, what seems to be quite extraordinary, the speed of light (denoted by c) is constant, independend
of any frame of reference.

The last one is not so trivial. On a human scale, we know that if you are in a train, which
moves with a speed of 100 km/h with respect to the ground, and you are inside the train and shoot
an arrow with 100 km/h in the direction of movement, the speed of the arrow with respect to the ground
is 200 km/h. Likewise, if you drive a car with 70 km/h, and someone overtakes you with 72 km/h,
then for you the relative speed of the other car is only 2 km/h.

It's always simply a direct (vector) addition/subtraction of speeds (velocities).

Fig 2: Illustration of 2 frames of reference S and S', moving with constant speed.



Figure 2 illustrates this. An observer in S might think that he is stationary. Frame S' goes by,
with a speed of 20 m/s in the +x direction, relative to frame S.
Ofcourse, an observer in S' might think that it is he which is stationary, and that it is frame S
which moving in the -x direction with a speed of 20 m/s.
Let's return to the observer in S. If the observer in S' shoots an arrow with 30 m/s in the +x
direction (relative to S'), then the observer in S measures the speed of that arrow to be 50 m/s.

If you would replace the arrow, with any form of electromagnetic radiation, like
radiowave, radar, light etc.., then all observers, no matter which frame of reference, would
measure the same constant speed, namely the speed of light, which is universally constant.
This is highly remarkable, and will have profound implications to the structure of SpaceTime,
as seen by different observers in different frames of reference (different in the sense of speed
in some direction, like the x-axis).

In many articles, the speed of light ("c") is a central theme. However, visible light is just one of
the infinite manifestations of ElectroMagnetic (EM) radiation, which has an infinite spectrum
of frequencies (and energies).

So if you are in S', travelling with with 30% of "c" with respect to S, and you turn on a laser pointing in the +x direction,
then observers in S and S', still will only measure the same constant speed of light (denoted by "c").

The following is not an adequate solution to the riddle of the constant speed of light.
There is a relation that couples "c" to 2 fundamental electrical and magnetic constants of the Vacuum, namely
ε0 and μ0 which represents the "vacuum permittivity", or "permittivity of free space".
These constants say "something" about the capability/ability of the vacuum to permit electric- and magnetic fields.

c2   =    1
----
ε0 μ0

Viewed this way, and assuming ε0 and μ0 are constant throughout the Vacuum,
then c is constant too. Again, this is not adequate enough, as a full explanation as to why c is constant
in all frames of reference.

Let's go to the findings of Einstein in STR.

Suppose we have frames S and S' again. Suppose you are in S, which you think is stationary.
Ofcourse, you can specify Space coordinates in your frame, using (x,y,z). The time in your frame
is denoted by "t". While the x, y and z can vary ofcourse, you assume that t is the same throughout
your frame of reference. That's no more than a valid assumption.
Suppose you are located in the Origin of frame S, that is (0,0,0).

However, frame S' moves with speed "v" towards the +x direction, with respect to the (stationary) frame S.
An observer in S', uses the Spatial coordinates (x',y',z'), and time t'.

From a classical point of view, the times t and t' are exactly equal. This is also in correspondence
with all experiences in human life. The time in a plane is exactly equal to the time on the ground.
This is not exactly so in STR. However, the discrepancies will get clearer as v gets closer to c.
For now, we denote the time in S as t, and the time in S' by t' (although classically, they should be equal).

Classically, an observer in S' would say that the coordinates of S and S' relate in the following way:

x' = x - vt
y' = y
z' = z
t' = t

Since the relative movement of S and S' is only along the x-axis, it follows that y=y', and z=z'.

The set of equations above, is often referred to as a "Galilean Transformation".

Einstein futher reasoned in the following way. If a light explosion would take place, then the spherical
wavefront would be seen as equal by all observers in any moving frame of reference.
It means for our observers in S and S', that:

Sperical wavefront described from S:

x2 + y2 + z2 = (ct)2 = c2t2    

We can describe the sperical wavefront from the perspective of S' too. Then it will be:

Sperical wavefront described from S':

x'2 + y'2 + z'2 = (ct')2 = c2t'2    

Both equations describe the same "distance" in Minkowski SpaceTime.

Thus:

x'2 + y'2 + z'2 - c2t'2 =d

x2 + y2 + z2 - c2t2 =d

But S' is moving into the +x direction only (as viewed from S). There is no reason
to expect "any effect" along the y and z directions. Sure, as you will see in a few minutes,
in the dimension in which we indeed have a "speed" ("x"), we will see a large effect.
But in the transpose directions, thus in this case the directions "y" and "z", there is no effect at all.
It's still reasonable to say that:

y' = y
z' = z

The distances in Minkowski spacetime as showed above, then reduce to:

x2 = c2t2    (1)

x'2 = c2t'2     (2)

Thus:

c2t'2 - x'2 = c2t2 - x2     (3)

This is still the metric as we should use in Minkowski SpaceTime, but we were able to eliminate
the "y" and "z" coordinates.
Since (1) and (2), are the same distance in Minkowski SpaceTime, we were able to write down (3).

These equations can be solved, that is, express x' in terms of x and t, and express t' in terms
of x and t.

The math is not too hard, but a little too spaceous to write down here.
You can take a look at one of my earlier notes, which is says a little more on STR,
and indeed shows the derivation of the solutions.

If you are interested, then you might want to take a look at this note.

Below you will see the solutions for x' and t'. These are the famous "length contraction",
and "time dilation". It starts to "live" if you really see an example. That will be done below.
For now, let's first present the solutions for x' and t':
____________________________________________
x'  =    x-vt
-------------
√(1-v2/c2)

y'  =  y

z'  =  z

t'  =    t - (v/c2).x
-------------
√(1-v2/c2)
     
____________________________________________

Do you see that, for example t', is dependent on the speed "v" of S'?
From a classical viewpoint, that's absurd. However, from de deductions of Einstein,
it's really true. It simply means that the clocks in S and S', run at a different rates.
An observer in S, will see that the clock in S' runs slower.
When you see a simple example, these conclusions will start to "live".

The solutions of Einstein, as presented above, simply were possible by postulating
that "c" is constant in any from of reference, which already is "unclassical" by itself.

If we want, we can simplify the equations above, if we use the "gamma factor" γ, which is:

γ  =        1
-------------
√(1-v2/c2)
     

In many articles, however, folks call γ = √(1-v2/c2), "the Lorentz factor".

Since the γ factor is common among the transformation equations,
we may also write (for v along the x-direction):

____________________________________________
x'  =   γ  (x-vt)

y'  =  y

z'  =  z

t'  =   γ  (t - (v/c2).x)
     
____________________________________________


The equations above, are called the "Lorentz Transformations" (for "v" along the x-direction).
Note that the " γ factor", to a high degree, determines the relativistic effect here.

Take a look at the first equation for x'. Note that if v is ver low, then √(1-v2/c2) is practiclly "1".
Thus it means that the equations converges to the Galilean Transformations for low speeds.

Examples:

Example 1:

Suppose in S, we have a marked segment L0 = 1m, as a segment along the x-axis.
Suppose further, that frame S' is in rest too, just as S is, and they perfectly coincide.
In S', we have the same marked segment L', thus it has a length of 1m too. L0 and L' coincide too.

Now, suppose that "suddenly", S' moves with a constant speed of 0.7c along the +x direction.

What does the stationary observer in S measures of L', when S' moves with that speed?

Answer:

L' = √(1- 0.7c2/c2) L0 = 0.714 * 1 = 0.714 m

So, according to the observer in S, the L' has shrinked. In other words, the spatial dimension
along the direction of movement seems to be contracted.

Note that in this example, the speed "v" was extremely large. It's 70% of the speed of light,
which is extremely fast indeed. True, relativistic phenomena will show better if the speed
of the moving frame of reference is significantly above 0.1 c.

Example 2:

Suppose in S, we have L0 = 1m.
When S' is in rest, we have the same distance L = 1m.

Now S' moves along the x direction with (only) 1000 km/s, which is about 0.003c.

How long does a stationary observer in S, measures L', when S' moves with that speed?

Answer:

L' = √(1- 0.003c2/c2) L0 = (practically) √1 * 1 = 1 m

With low speeds, say below 0.01 c, relativistic phenomena are hardly observed.
That's why classical Newtonian mechanics works great with speeds that are only small fractions of "c".

Indeed, with speeds below 0.01 (where c is about 300000 km/s), the "world" looks fully classical again,
and that's why on a human scale, classical Mechanics still works fine.

"Length contraction", and "time dilation", have been experimentally confirmed at incredable precision.
For example, a clock on a sattelite runs a bit slower, exactly as predicted by the theory.
As another example, the decay rate of some elementary particles is longer, when they move with high velocity,
compared to Lab conditions.

This seems like a strange "flexibiliy" of Space. However, in SpaceTime (x,y,z,ct), it follows naturally
if the speed of "c" is constant in any moving frame of reference.

Ofcourse, the material above, evidently, only represents just a tiny glimpse on "The Theory of Special Relativity".

1.4 The essential meaning: Lorentz symmetry and SpaceTime distance:

Above, we already have seen an example of the Lorentz metrc (distance) in Minkowki SpaceTime.

ds2 = -c2t2 + dx2 + dy2 + dz2

The minus sign in "-c2t2" was not explained well above, but I can tell you
that the "extra" coordinate "ct", in fact should be "ict" (Henry Poincare, 1905) where "i" is the imaginary number
from Complex number theory. If you square that, it will give rise to the "-" sign.
I don't think that the very details are very important to the discussion I like to present.

To let the equation above, resemble more to a "distance", or interval "Δs", we can rewrite it like:

Δs2 = Δx2 + Δy2 + Δz2 -c2Δt2

where Δ is a universally accepted symbol for "small part", instead of infinitesemal qualifiers.

The equation means that the Lorentz distance (or Minkowski distance) between two "events" in SpaceTime,
is constant.

Since we speak of SpaceTime (Space, Time), points are better qualified by events (physical events),
that may take place, the one later than the other. It's possible to connect these events by light.
Suppose one particle (particle 1) emits a γ photon, which may be absorbed by another
particle (particle 2) somewhere else in SpaceTime.

Since the distance in SpaceTime is constant, you may sneaky contract a spatial component (say x), but them
the clock must run slower in order to have the same distance between the two events, again.

Using that as a principle, then apply some math, you will get the Lorentz transformations as
listed above.

You also might see that this framework enforces causality. It's not possible in this model, that,
for example, a particle 2 absorbs a γ photon, before it was send by particle 1 in the first place.

1.5 Lorentz violations:

In the above, we considered two frames of regerence, S and S', where S' had a constant velocity
along the +x direction., relative to S.
Ofcourse, we could have also choosen for a movement of S' along the -x direction, or along
the y-axis, or along the z-axis, or actually any direction in the coordinate system..

It would not have changed anything fundamentally.
The Lorentz transformations would still be the same format.

In SRT, there is no preffered direction in SpaceTime, and no dependency on whatever coordinate
system is used. This is also called "Lorentz symmetry".

Note:

You will see later that physicists appreciate (or nearly demand) that a "concept" is rotationally invariant,
invariant for transformatons, invariant for phase shifts, and invariant for change of coordinate systems.
This holds all the more for "something" that might be called a "fundamental concept".


This symmetry, or "gauge invariance" is reflected in theories which are (sort of) written or re-written
using the Yang-Mills fundamentals (or idea's).

Some theoretical considerations....

-(1):

Is it really true that there exist no preferred "something" in SpaceTime?

Maybe there exists an extremely small bias towards some "direction", or energy potential
in the Vacuum, or "hidden" yet undetected field in the Vacuum, or even location in the Universe,
or even location in our own local Milky way etc.. etc..

It's difficult to say something truly useful on the above speculations.

But there are some anti-symmetrical things indeed.

If you would observe some special physical systems, with some particles having electrical charge, and spins,
and "invert" the charges (so that + will get -, and the other way around), or mirror them (in an actual mirror),
then sometimes surprising effects can be witnessed: Violations of symmetry.

There are some fundamental forces in our world, like the Strong nuclear force, gravity etc..,
but something called the "weak interaction" displays, as many physicists believe, some un-symmetrical
"behaviour" indeed.

Contemplating using this sort of intel, and the principles of STR, still have not resulted
in very clear statements.

-(2):

Einsteins STR, uses a continuum, flat (not curved) 4D SpaceTime.

But what if the quantization of "Space" is true? Then, using the theory above, when a frame of referece
is (almost) infinitesmal close to "c", the quantization of Space must be "felt" in some way.
You can go very far in "length contraction", but what if when you come so close to the scale,
where Space quanta cannot be ignored anymore?

This sort theoretical considerations have also led to the search of "Lorentz violations".
Many experiments have been performed, to a very high prescision already, but no anomalies
have been detected yet.

The hope is, that measurments of any possible violation, might produce some insight
to which of the competing "Quantum Gravity" theories, is best.

1.6 A few words on General Relativity:

General Relativity is too much of a grande Theory, to discuss in any value in such a note like this one.
However, it is possible to distill a few main points.

General Description:

Einstein's Theory of General Relativity, is much more involved than Special Relativity.
One reason why it is called general, is because accelerated frames of reference are studied,
instead of "only" frames of reference moving with a uniform velocity.
In effect, all sorts of relative movements are considered.

One astounding finding was, that "gravity" is equivalent to acceleration.
The acceleration is then due to curved SpaceTime.
This was absolutely completely different from the common classical view, before 1916,
where gravity is a Force, just like the electrical- or other know forces.

The "core" idea of GTR, is that Einstein came up with the theory that SpaceTime is a
geometric object whose curvature is determined by the distribution of energy and matter.
The curvature determines how free objects will move in that curved SpaceTime.

Thus gravitational force is no longer a force in the classical Newtonian sense, but a mere
manifestation of the curvature of spacetime.

In a type of math, which was later called "differential geometry", curvatures of spaces (manifolds)
were already explored by Gauss, Riemann, Christoffel, Cauchy, and too many other mathematicians to name here.
For some important theorems in that realm, we can go back to the years around 1850, or even earlier.
Indeed incredable, that this mathematical branch needed well over 100 years to develop into a mature framework
where it is still intensely used by physicists today.

But Einstein too, relied heavily on "differential geometry" in the period he developed GR, from 1905 - 1016.

If you would consider some "manifold", like some 2D surface in 3D, it's possible to introduce
a tangent vectorfield "along" that surface, which describes the "rate of change" of how that surface
actually bends. It's a simple example, which hopefully you can visualize in Space.

An extension to a vectorfield, is a description using a tensor object. This mathematical object,
makes it possible to "express" more twists, in multiple directions, in any point.

An example of one of the field equations in GR:

A tensor is a very suitable mathemathical object to capture the differences in twists and bends,
from a point to other neighbouring points.
It's therefore no wonder Einstein found a way to describe the curvature in SpaceTime, using
implementations of tensor objects.

This can be illustratred by one of his field equations, where Guv and Τuv are tensor objects:

Guv + guv Λ = 8 π Τuv

(where G=c=1, or geometrised/normalized units)

In the field equation above, the curvature of SpaceTime (Guv) is related to the mass-energy distribution
uv) which is present "in that neighbourhood".
It's absolutely remarkable, that this mathematical expression "links" mass-energy (or simply mass) to
curvature in SpaceTime.
It's a departure from classical Physics, where Gravity was considered to be a "force", just like
for example the Electric force.
But Einstein managed to link the curvature of SpaceTime, to mass-energy.

Now, if somehow it can be made plausible that a free object follows the curvature of SpaceTime,
then (maybe?) we are close to understanding how "mass/curved spacetime/path of an object",
all are connected by the Theory.

Why does a small free object follows the curvature in SpaceTime?

If you would think that it's a trivial question, then you must be a relative of Einstein.

If a particle is small, there is hardly any "feedback" into the "warped" SpacetIime, which itself is due
to some larger mass distribution "nearby".
So, a small test particle, "in some way", finds it's path in curved SpaceTime. So, what is the path here?

If we would not consider a small object, then this object itself would significantly warp SpaceTime too,
which is covered by Einstein's GTR, but it's very complex.
It does not have to be really as small object, as long as it's small relative to the mass that curves SpaceTime
in the first place. It's a bit similar to Earth orbiting the warped spacetime due to the Sun.
The Sun is immensly more massive than Earth.

Short definition: In differential geometry, a "geodesic" is a generalization of the notion of a "straight line"
in "curved spaces".


Now, the question thus equivalent to:

The motion of a small test particle, is completely determined by the bending of the SpaceTime.

Some folks can prove it, by using the equivalence of intertial mass and passive gravitational mass.
These two interpretations of "mass" has not mentioined at all, in this simple text.

Others can prove it by using the general equation of motion in curved SpaceTime.

It's not so very trivial. One idea is using the concept of parallel transport. You can consider a tangent vector
along the motion, or orthogonal to the motion. The motion is in curved SpaceTime, ofcourse.
If the orientation of that vector does not change relative to the path of motion, then you stay on the geodesic.

If you are on a curved sphere (a surface) in R3, and you hold a stick exactly in front of you, and
you walk along a "great-circle" (a geodesic), the orientation of the stick (tangent vector) does not change.
So, if you go from the equator to the Northpole, and keep on going the straight line (the great-circle), the
tangent vector does not change. However, while on the Northpole, and you change suddenly direction, like turning left,
and then go back to the equator again, then there was a rather sudden disruption in the orientation
of the tangent vector. That does not correspond to the motion of a free particle moving in curved SpaceTime.

Relativity is a Theory using 4 dimensional SpaceTime:

Throughout section 1, it was hopefully clear that SpaceTime is 4-dimensional, which is reflected
for example in coordinates like (x,y,z,ct).

I like to stress that fact, since in section 3, Kaluza-Klein theory, which is a remarkable theory,
is an attempt to unify Einstein's GR, and the ElectroMagnetic (ElectroDynamics) Theory of Maxwell.
The arena where that seems to work, is a 5-dimensional SpaceTime, which is very remarkable.

The ideas in Kaluza-Klein, inspired many other Theories, even very modern ones.

However, Kaluza-Klein, does not seem to fit well enough in, e.g., modern Yang-Mills concepts, and even beside
that, Kaluza-Klein was more or less superseded by String-, M-, and Brane theories.

2. A few words on Planck's length, and Planck's time.

The "length of Planck", is an extremely small length, namely about 1.6 x 10-35 m.

Associated with this length, are two other values, namely "Plancks time", and "Plancks mass".
Of those two, "Plancks time" is somewhat more easy to understand, since it's the time needed for light to "traverse" Planck's length.

In order to get an appreciation on how small the "length of Planck" actually is, then take
a look at the following figures:

-The Bohr radius, that is the classical radius of the Hydrogen atom is about: 5.310-11 m.
-The classical radius of a proton is about: 0.87 10-15 m.

If we compare Planck's length to those upper examples, like the radius of a Hydrogen atom,
or what is often taken as the "classical" size of a proton then we will really appreciate
how insanely small Planck's length actually is.
If you would "inflate" a proton to the size of the Sun, relatively speaking, you still could not even see Planck's length.

This length is formed from other Universal constants (like the speed of light and others),
but we will also see on what theoretical basis this length was originally derived from.

We have to be very careful on how exactly to interpret such a small length.
At the same time, it cannot be denied that "Quantum Gravity" theories take Planck's length
as a reference point, that is, a scale that represents the dimensions of Space quanta (spins, loops etc..).

Planck's length is the following:

lp = √ (ħ G / c3) = (about) 1.6 x 10-35 m.

where c is the speed of light, ħ is the socalled reduced Plancks constant, and G is the universal gravitational constant.
So, the lenght of Planck is "build" from very fundamental constants from physics.

The theoretical time required for light to cross a distance of 1 Planck length, is about 5.4 x 10-44 seconds.

How is Planck's length derived? Where does it come from?

We are not going to do much math in this text. But basically, if one would compress one of those other constants, namely "Plancks mass"
to the "Schwarzschild radius", which is the critical radius of a Black Hole, then one would arrive to Planck's length.
To be honest, we would need to consider the Compton wavelenght as well, but we skip that here.

Note that Schwarzschild radius" is that metric, where SpaceTime fully collapses (into something we are not fully sure of).
Usually, the Schwarzschild radius can be understood as the "border" of a black hole.

By the way: some modern ideas in physics around black holes, will certainly be a subject in this modest note.

Some physicist tie the Planck scale to a phenomenon called Quantum Fluctuations, where Energy "pops up" from
the Vacuum in the form of a particle-antiparticle pair, which quickly destroy each other again.

Now we may see why "length of Planck" could be of significance of our discussion of the "Vacuum and SpaceTime".
Here are a few "suggestions":
  1. It's the lenght where possibly, all regular, smooth, continuous SpaceTime principles do not apply anymore.
  2. It's possibly the scale of SpaceTime quanta.
  3. It's the scale where a compressed Planck Mass (1.22 1019 GeV/c2) will collapse into a black hole.
  4. It's the length where possibly Quantum Mechanics and Gravity might unite in a single theory.
  5. It might be the most basic "container" of information in "Quantum Information Theory".
  6. It might be the characteristic length of "strings" in Superstring theory.
  7. It might be the characteristic length, related to "Quantum fluctuations" in the Vacuum.
I can simply list all that stuff above, but it then it simply just has to be illustrated, with some core concepts
of such theories. That is what I will try to do in the following chapters.

It's true that Physics is in full development, and a very definitive, complete, Theory is simply not present.

In the next sections, it's very important to give a quick overview on the fundamental themes that gradually,
found it's way into physics, like the Yang-Mill ideas, Gauge invariance, Quantum Mechanics, Quantum Field Theory,
the position of Relativity, Quantum Gravity, the Standard Model etc...

It's important to get a feel into that "stuff". Ofcourse, it will not be in depth, and I could not ever
cover it in depth, since it takes an incredable amount of knowledge, and thus an incredable amount of time to master.

In depth studies indeed takes years. But I am confident I am able to at least touch upon these subjects
in order to convey a feel for the fundamental ideas behind those themes.

However, what appeared shortly after General Relativity, namely the Kaluza-Klein theory (around 1921),
gives a certain perspective on SpaceTime and unification. In that sense it's important.
So, I like to do that first.
But not before a tiny bit of math has been put on the screen.

3. A tiny bit of Math.


4. Kaluza-Klein.