Version: 1.2

By: Albert van der Sel

Status: not ready yet

Remark: Please refresh the page to see any updates.

Fig 1:

A picture (left side) illustrating the (hypothetical) quantum "foam" of spacetime at extremely small distances.

The figure might be percieved as a bit

would believe that such a "structure" could represent the "smallest scale" of reality.

However, nowadays, more and more physicists are geared towards a "sort of" (discrete) knotted SpaceTime structure,

and many physicists who are working in the field of Quantum Gravity, have replaced the quantum "foam", by quantum "loops"

or "spin foam".

Mind you, a smaller set of (theoretical) physicists, are working on models that associate the smallest

space-time building blocks, with "black holes", or (socalled) ER=EPR Plank-scale "wormholes".

Other startling perspectives go around too:

The following is "just" theory ofcourse, however, it is based on real ideas, expressed by quite a few members in the

community of physicists. For example, the "holographic principle" uses elements from multiple theories,

like from Superstring Theory / M-Theory.

Some collaries suggest that we live in a "shadow" projection (just a manifold or subspace). This is so since

Universe, just simply is a "boundary" in that Superspace. So, all sorts of physics goes on in that Superspace,

while we only can observe events in our own 4 dimensional projection (our "universe", which is "just" a manifold,

or subspace, in that superspace).

The socalled "Brane world" cosmology, is highly linked to ideas such as the above.

The upper paragraph might be percieved as a bit

But, it's really physics (!) but way beyond the classical/traditional form.

Today, it's fair to say that "modern theories, or models" (like superstring, or quantum-gravity) still do not

fully decribe "the World", like for example "

What that "smallest" of stuff actually

in all chapters of this note.

It's really so that physicists try to make sense from good (semi-) classical Theories like Quantum Mechanics, and

General Relativity, to get to a point where some theory might apply in all Domains (with respect to distant scales, energy scales etc..).

Two rather theoretical (and partial) succesful attempts are "quantum-gravity" and String theories.

Let's try to take a look at some "properties" of the Vacuum and SpaceTime,

as many physicists think as it might be, according to some current theories.

So, here we will discuss a few those insights.

However, it's not always sure how "well" such ideas connects to "reality", but anyway..., it's fun...(I hope).

Although I believe the stuff below is OK, it's probably not good to jump to conclusions. It's better to build

a view "step by step" (or "chapter by chapter"). However, this will be a supersimple text.

Right now, I am not sure, but maybe the following (supershort) sections might be of interest (list may change):

2. Virtual particles, Casimir effect, ZPE, and all that.

3. The Dirac Sea, and similar ideas.

4. Einsteins relativity principles of SpaceTime.

5. Some info on Theories on Black Holes and Wormholes.

6. Vacuum Energy.

7. Some ideas of Discrete Space, Spin networks, and Loop Quantum Gravity.

8. Quintessence, Dark Energy, or Cosmological constant: accelerated expansion of SpaceTime?

9. The Standard model, and "Gauge" and symmetry formalisms.

10. Superstring Theories, and Brane world scenario's.

11. The Holographic Principle.

12. Other stuff.

Notes:

(1): I also want to say that this note classifies under "fun stuff", meaning that it is serious,

but the main objective is that it should be fun to read....

There is very little math, and if it's present, it usually is some "shortcut" (hopefully a clever one),

that still presents the "keypoints", but is not (too) boring (I hope)....

I have no idea if such a text like this one, could contribute to a broadening of vision on the subjects. But I do hope so.

(2): The first few chapters might be percieved as "

are often presented as true independend entities. Well, some modern ideas view that somewhat differently.

Don't worry: a highlevel introduction of modern ideas is certainly one of my aims of this note.

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.

If we compare Planck's length to other very small examples, like the

10

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

We have to be very careful on how exactly

but there is no true solid basis for very exact statements.

Planck's length is the following: l

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.

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

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).

Now we may see why "length of Planck" could be of significance of our discussion of the "Vacuum and SpaceTime".

It's the lenght where possibly, all regular SpaceTime principles do not apply anymore.

Although figure 1 above, might be a bit too suggestive and too speculative, the text above does explain to a certain extend

why people speak of SpaceTime "foam" at the dimensions of Planck's length.

(When we have talked about virtual particles (at a later section), this "foam" is often called "quantum foam".)

It's the same as looking at a pond. From a great height, like from an airplane, the pond looks smooth and calm.

However, when you are walking around the shore of that pond, you see all the waves and wrinkles in the water.

I'am certainly not suggesting that it's a "chaos" at such small scales, but if indeed there is significance in Planck's length,

then it's fair to say that it's still largely unknown territory, but then it is an important factor on how we must view

the Vacuum and SpaceTime.

dimensions are used in those theories. Then those dimensions have a metric like the Planck's length which makes them completely

unnoticable.

Also, Planck's length is likely to be important for "Quantum Gravity" theory too. It might be true that Quantum Mechanics,

and Gravity, gets "unified" at scales or Energies comparable to the Planck length.

It's noteworthy that especially "Loop Quantum Gravity" succeeded (as many say) in describing SpaceTime

in a "discrete" quantum manner. Planck's length then, is the "distance" where loops or spinfoam is measured in.

Could we ever "probe" such small lengths? To reach a cross-section comparable to Planck's length, for example with a

particle accelerator, we need an energy that would need to be trillions and trillions of times more, than our

presently largest accelerator can produce.

that "special relativity" teaches us that in a moving frame of reference, Lorenz transformations will lead

to "length contraction" (in the direction of movement) as observed by another observer in another frame of reference.

Then... what happens to the length of Planck? The question still leads to interesting studies.

In general however, folks consider "special relativity" as in relation to 4 Dim Minkowski space, and when

curved SpaceTime of General relativity is considered as well, "things" changes a bit.

However, the puzzle is not solved (if it indeed is a puzzle).

Since you have read in section 1.1, that the length of Planck is closely related to the Horizon (Schwarzschild radius),

at which the

special relativity into account, do not rule out that SpaceTime is a strange structure of Black Holes at the most

tiniest scale.

It's possible that a better interpretation arises, if we also consider string theories. The lenght of the basis state

of a superstring is likely to be in the range of the length of Planck. In this case, we have a fundamental "lenght"

associated with an elementary particle, in it's basis state, and it might be meaningless to ask questions

below that scale. Another thing might be considered too. If Heisenberg uncertainty relations still hold,

then we probably cannot say anything usefull (at or) below the Planck scale.

General Consensus (?):

The length of Planck is reckonized to be of significance. However, it's not fully clear to what extend that is.

Generally speaking, it is felt that by unifying

on the importance of these constants.

With respect to SpaceTime foam (or possibly Quantum Foam), fluctuations of the Vacuum is not disputed, however, the breakdown

of time and space on the smallest scales, is doubted by most (or at least many) physicists.

For now, Large scale (astronomical) observations do not support any sort of discontinuity of SpaceTime over long distances.

Contrary, theoretical physicists in the fields of Superstring- and Brane theories use Planck's length as a natural distance

as to where "hidden" or "extra" dimensions (> 3+1), are curled up, and that plays a vital rol in those theories.

Also, especially in

is treated as a fundamental entity, and by some, even viewed as the

Some (theoretical) physicists, are working on models that associate the smallest

space-time building blocks, with socalled ER=EPR Plank-scale "wormholes". In many cases, they focus especially

on the Quantum Mechanical effect of "entanglement", and believe it or not, some articles make sense how such planck scale

wormholes connect entangled particles. However, many physicists are sceptical, and like much more "body" on those models.

Please see chapter 6 for more information.

We are not done yet with those amazing values, but let's first explore some other properties of the Vacuum and SpaceTime.

Step by step we are going to build a view on SpaceTime, as many physicists sees it.

other means in physics. Some other physicists see it quite differently.

Here, I will try to provide a (somewhat) wider (but non-math) view on virtual particles.

If you consider the term "virtual particle", then indeed it does not really "sound", as if we are talking about

"real" (permanent) particles, like for example electrons or quarks are. Indeed, why else would we use the prefix "virtual"?

Yes, you can find people who say they are extremely short-lived "particles", popping out of the Vacuum at times, and others

who think that they are only "disturbances" of some field.

Still others, simply see them as mathematical constructs that helps explain particle interactions, like in Feynman diagrams.

According to some folks, it's even best to describe (or talk) about "virtual particles" in the context of "measurable",

or observable, "effects" only. That is:

This is motivated by the fact that sometimes virtual particles seems to be able to temporarily "escape" from physical laws,

for a brief moment, as long as in the end

I would like to start by saying that quite a few different angles are possible to talk about "virtual particles", and probably none of them

is "perfect". But having seen some of those different approaches, may help in appreciating the phenomenon of "virtual particles".

Let's just simply consider a few cases:

Let's take a look at a "boson" (force carrier) of some sort, like the "W

These are the mediators of the "electroweak" interaction. Now, it has been detected in collision experiments, using particle accelerators.

In that sense, it really seems

we do not observe it "just like that". It will just temporarily "exist", if "electroweak" interactions takes place.

But are they good examples of virtual particles? No, not really. Althoug they only have a half-live of 3 x 10

and thus only exist for a very brief moment, they have rest mass, and other observable properties of real particles.

It's not really easy. The "W

will not put them in the virtual particle category.

In Quantum Mechanics, one famous result is the

It roughly says that two non-commuting operators (observables) of a quantum system cannot both have zero uncertainty

at the "actual" values of the associated observables.

One famous relation, states that the

In formula: Δp.Δx ≥ ħ/2 (where p is the momentum, and x is the position).

It means that the more certain you are in the momentum of a particle, the uncertainty in position will increase, and vice versa.

Yes, nature seems to use a "rule" that if one observable gets more precise, the other one gets more fuzzy.

A similar relation holds for "Energy" and "Time": ΔE.Δt ≥ ħ/2 (where E is Energy, and T is time).

Please note that the inequality uses "≥ ħ/2". In a way, this almost makes it mandatory that ΔE energy pops up from

the Vacuum, for a certain duration Δt.

If Nature indeed works in the following way, that energy is "borrowed" from the Vacuum, for a certain timeslot, and produces a pair

of "virtual" particles, and within that timeslot, they annihilate each other again, then the "bookkeeping" is always in order.

Now this example (example 2), really shows us virtual particles.

Those two examples are not perfect, but they show some important features of virtual particles.

The phenomenon of example 2, is also sometimes called

It seems that the "Quantum fluctuations" is a synonym for the contineous production and annihilation of particle pairs.

In principle, the pair of particles can be of any type, as long as they are able to annihilate each other again, like

for example a virtual electron-positron pair.

The following is not

Virtual particles can be understood to be a manifestation of the time-energy uncertainty principle. That would be a

physical interpretation. Theoretically, in some disciplines of physics (QFT), a particle existence is understood to be goverened

by creation- and annihilation operators.

In case of virtual particles, these operators work not the same way as in the case of "real" particles, and thus for their

"existence", a probability distribution is in effect. Hence, the "strange" popping up and disapperance of those virtual particles.

Note:

The term "Virtual particles" is not synonymous to "Quantum fluctuations". However, "Quantum fluctuations" produces Virtual particles.

Virtual particles might for example be the "electron/positron" pair, but "virtual photons" might be placed in that category too.

and other effects, might be more difficult to understand (not everbody would agree on this).

However, especially for the Casimir effect, also alternative origins have been theoretically explored, as an alternative

for quantum fluctuations.

If the vacuum produces short-lived charched "virtual" particle-antiparticle pairs, then they would on average

In a way, the vacuum is in effect counteracting the field, or charged particle.

This effect is called

Has this been observed in experiments? In several ways, yes.

For example, the small "Lamb" shift

in spectral lines of Hydrogen (and other elements), is attributed to the interaction with the vacuum.

Similar to the former effect, vacuum polarization is very likely to be experimentally observed in muonic atoms.

Some physicists have thus some "worries" on making statements on "bare" (or naked) charged elementary particles.

Imagine the following setup:

Two uncharged small metallic plates are placed in a vacuum, parallel to each other.

The distance between those plates is extremely small.

What can be observed, is a small attractive force between those plates.

What is considered to be a resonable explanation for this effect, is the following.

Both plates exists ofcourse in a "sea" of virtual particles. According to Quantum Mechanics, particles can be described

as waves, and that holds for our virtual particles too. The "number" of different wavelengths that "fits" between the plates

is smaller than on the "outside". Hence, a net force exists, pushing the plates towards each other.

From an "energy" viewpoint, the Vacuum then performs "work", and that's one motivation why some folks talk about

intrinsic energy in the Vacuum. This, while the term "ZPE" means "zero point energy", which hints at "0" energy.

If you think that's a bit "strange", you are probably right.

Some have doubts that quantum fluctuations are actually the cause of the observations.

Although "quantum fluctuations" are not really disputed, it is argued that the main cause for the observed effects

are forces due to relativistic effects on internal charges.

There are several articles describing this alternative interpretation.

If it is ultimately found that this alternative approach is better, it just has to have a certain impact

on thoughts about "ZPE".

General Consensus (?):

Virtual particles do not exist in the sense real particles do. They are often regarded as helpful mathematical

constructs. Also, the need to be able to

to label them as "real".

At the same time, some argue that in (e.g.) pair forming and other physical events, a partner particle of a virtual pair

may "become real", and thus (according to some) the status of "Virtual particles" stay in a "grey area" (for now).

Also, the theory of creation/annihilation operators, although different for virtual particles, makes them more "real"

compared to other theories.

Many physicists believe that the "time-energy uncertainty principle" is the physical basis for Virtual particles to

"exist" for a brief moment.

Amzazingly, it is consistent with both the principles of quantum mechanics and the theory of special relativity.

We are not into math in this text, but believe me, Dirac's work is rather involved.

Actually, the equation and further formulations, form a "frame work", which up to this day, must be considered

to be a great piece of work and insights.

By the way, Dirac was very important for the development of Quantum Mechanics, where, among many other stuff,

he introduced a "vector" (bra ket) calculus, which was a large stimulus for QM.

He also predicted the "positron", which is the anti-matter partner of the electron. In a minute, we will see why

Dirac came up with the positron.

Additionally, Dirac used the "extended"

which, if the momentum "p" would be "0", goes to Einstein's famous relation E=mc

(By the way, it's Einstein formula. Using E=mc

has to do with in which frame of reference you observe a particle. In rest, p=0, and the extended formula goes to E=mc

However, from "E

E

For that time, such an idea was completely new (a bit radical actually). Furthermore, Dirac reasoned along the following lines.

There are many occasions where an electron pushes off energy, so, in many occasions it should go into a negative energy state.

However, these are not observed.

To solve this, Dirac assumed that "a negative sea" of electrons existed in the Vacuum, where all possible

quantum states were already occupied. Hence, an electron with positive energy can't "go" into the negative sea.

This is so, since fermions obey the socalled Pauli exclusion principle.

This "sea" is called "The Dirac Sea". However, Dirac went further. What if a an electron with negative energy gets "exited"

and goes into positive Energy? Then cleary, "a hole" remains in the negative sea, which Dirac (later) called a positron.

(at first, Dirac believed that the hole was the proton).

Not much later, the existence of the positron was indeed confirmed.

Such ideas also led to the "Hole theory" of the Vacuum.

Fig. 2: illustrating the "Hole theory", with an electron (positive energy) and a positron (negative energy).

How do physicists today view Dirac's ideas?

into "Quantum Field Theory" (QFT).

The solutions for particles with negative energies, are today reinterpreted as

So, although the power of Dirac's work was, and is not disputed, a newer an broader theory (QFT) was developed.

But even in those days, a seemingly infinite see of electrons in a negative Energy state, was never much

appealing to most physicists at that time, but most reckognized the implicit power of "the concept".

Today we can find many articles that try to unite the "hole theory" concept with bosons, for example:

Dirac Sea and Hole Theory for Bosons I -- A new formulation of quantum field theories

Above is just an example, but it shows that the "hole theory" stays sort of vivid for researchers.

Another sort of "follow up" is a brand new theory called "causal fermion systems". I myself are now trying to

understand the significance of this theory, and hopefully, that will work out.

One key point is that all known aspects of quantum mechanics are actually "encoded" in a causal fermion systems.

If you like a non-technical view (but still very technical and involved), you might like the following article:

Causal Fermion Systems as a Candidate for a Unified Physical Theory.

severily crippled. So, that's what we are going to do in this section. Needless to say that my discussion here

is totally inadequate. However, listing some "keypoints" should be possible.

I absolutely

with respect to the great works of Einstein.

Einstein developed two main Theories: "The Special Theory of Relativity" (STR) and "The General Theory of Relativity" (GTR).

- STR is mainly involved in "frames of reference" which move with uniform/constant velocity with respect to each other.

The "curvature" of SpaceTime is neglected (or neglectible).

- GTR is "general" since it also involves "accelerations", and shows that "gravity" is the result of curvature of SpaceTime,

due to mass.

Due to the works of Lorentz, and especially Einstein (and at a later time also Minkowski), it became more and more clear

that an interwoven 4 dimensional

Before this was taken place, "classical" (Newtonian) mechanics, and electrodynamics, were extremely succesful in describing

physical events. However, as Einstein showed us, this is mainly due to the fact that in our human world, the "speed" of

physical objects is very low compared to the speed of light. At low speeds, the relativistic effects are hardly noticable.

Visible light, is just one specific sort of "ElectroMagnetic" radiation in general. Einstein took as a fundanmental principle,

that the speed of light

This fact may sound quite strange, and not in accordance to everyday live experiences. For example, if you are in a train,

which rides 100km/h, and you walk up front the with 5km/h, then relative to the ground your speed is 105 km/h.

Also, if you walk to the back with 5km/h, then relative to the ground your speed is 95 km/h.

Among a number of"assumptions", as the most important ones, Einstein took the following:

- The law of physics are the same in any frame of reference.
- Space (or SpaceTime) has no preferred "direction", thus we can perform an experiment along the x direction,

or y direction, or any other direction (like the direction where a frame of reference is moving to). - The speed of "c" in vacuum is independent of the speed of any frame of reference.

Now, let's "review" a few simple classical results first, before we go to "the real stuff":

1. Suppose you have a constant speed of 100 m/s (uniform motion). After travelling for 12 seconds,

the distance covered is 12 x 100 = 1200 m. After 20 seconds, the distance covered is 20 x 100 = 2000 m.

So, for a uniform motion holds: "distance covered = speed x time", or

2. If, in a coordinate system, you have a position as (x,y,z), but only move in the "x direction",

then we may say for the new point (x',y',z') where you arive in "t" seconds: x'=vt, y'=y, z'=z.

Since you only moved along the x-axis, then y and z did not changed at all.

3. Just to be sure you understand this: a sphere in 3D, with it's centre in the origin of a coordinate system,

and with a radius of "r", can be described with x

An unformly expanding sphere, with an expanding speed "v", can be described by at any time "t", by:

x

Fig. 3: illustrating the "classical" Galilean transformation.

I think we can understand figure 3 immediately. You see a frame of reference S', moving to the left (along x-axis) with

a uniform velocity "v". We also see a frame of reference S, which is stationary. As a side note, some elementary

relativity applies even here too. An observer in S', may rightfully think that he is stationary, and that S is

moving to the right with velocity "v".

Indeed: an "absolute" frame of reference does not exist.

In order to relate the coordinates between S and S', the classical Galilean transformation would work.

So, suppose we view S as stationary, and S' moving with a uniform velocity in the -x direction, we would have:

x' = x-vt

y' = y

z' = z

t' = t

And indeed, t'= t, since in the classical view, clocks tick exactly the same way in any frame of reference,

and time does not depend on velocity. This generally is what we as humans feel as OK, in our moderate

macroscopic World.

Now, are you ready to go?

Einstein reasoned along the following lines. If c is constant everywhere, independed in whatever frame of reference

you are, then a expanding spherical light "bulb" will be observed in exactly the same way, no matter what

your velocity is, compared to any other coordinate system, with whatever uniform speed.

So, he argued:

In frame S we see:

x'= |
x-vt ------------- (5) √(1-v ^{2}/c^{2}) |

t'= |
t - (v/c^{2}).x------------- (6) √(1-v ^{2}/c^{2}) |

(We already argued that y,y' and z,z' can be left out from the discussion. They remain the same.)

So, the critical thing is this: if relation (4) must hold (

relations (5) and (6) into (4),

I minimised the use of math here, and I sort of reasoned the "other way around", compared to technical

articles, but I don't care too much: It's not so relevant. What is most important here, is whether

followed the arguement here. If so: succes !!!! If not:

Now some startling conclusions can be derived from the equations (5) and (6).

If you are in frame S, then you observe shorter distances in S', and the time in S' runs slower too.

This is so, since the following is true. Let's first focus on "lenghts" along the x-axis in S '.

In S', any lenght is just the distance between two points x'

For both points, relation (5) holds. Now, the critical factor in relation (5) is:

1 ------------- (7) √(1-v ^{2}/c^{2}) |

If the velocity "v" with which S' moves relative to you (in S), compared to "c", is very slow then

√(1-v

However, suppose S' moves a large velocity, like 0.1 c, or 0.5 c etc.., then the factor becomes really relevant

and you will see that any lenght along the x-axis in frame S' has becomee substantially shorter.

If you know that the speed of c is almost 300000 km/s, or 3 x 10

for yourself (or just scout the Internet for some nice examples on "length contraction" or "time dillation").

After rearranging some terms, if you would have measured a length "L

was in rest (relative to you in S).

Then the "new" lenght "L" in S', if it now moves with velocity "v", would be:

L= L

So, for example, when S' was in rest, and you and an observer in S' marked a length of 1m, and S' then

was speeding with 0.5c, then L is about 0.86 m.

Why is this strange effect real? What's behind it? Why does seem SpaceTime to be so "flexible"?

The best explanation sofar, is the acceleration that has preceded, before S' reached speed "v".

Ofcourse, we need to explain that further.

For now, we know that a system moving with velocity "v" relative to you, then in that system lengths are shorter

and time runs slower. In low velocties, say up to 1000 km/s, the effects are very small (but measurable).

With low velocities, it all "looks" classical, and you must do your best to notice any effect.

With higher velocities, like 0.1 c (30000km/s), or even 0.5 c (150000 km/s), the effects are very relevant.

Equation (7) is often called "the gamma factor".

This factor often plays a key role in how properties change when things get relativistic.

So, in articles about STR, whenever you see "γ" in equations, you know it's actually a shortcut

for the formula of equation (7).

γ = | 1 ------------- (7) √(1-v ^{2}/c^{2}) |

In STR, in a frame of reference S' moving with speed "v" relative to frame S, "mass" will have increased too.

So, if you are in S, and observe S', then lengths will be contracted, the clock will run slower,

and mass will have increased too.

That should not amaze us too much. Eistein already showed us that "E=mc

when a frame of reference has a high speed relative to us, it's Energy is high as well, and so will be the "mass".

The different "characterizations" of mass, is not entirely without controversy.

Anyway, all physicists agrees on the fact that an elementary particle has a "rest mass" m

which can be observed when it is in rest. Now, suppose that particle now zips by with a high velocity, then

some physicists speak of a "relativistic" mass "m", which is larger than m

Most will agree that m = γ m

I will leave the mass stuff alone for now. Anyway, "mass" is more relevant in General relativity, so let's

revisit mass at that section.

In SpaceTime, you may denote points by (x,y,z,ct), so we can say that Einstein's SpaceTime is 4 dimensional alright.

Sometimes, such a point is denoted by (ct,x,y,z), but that's only cosmetic.

Often, such points are also named "events", as to place more accent on the fact that "some" event may happen,

which we may "follow" through SpaceTime.

We have seen some special things in STR, like that lenghts shortens, and that time slows down.

However, an "effect" will never precede it's cause. In short: causality is preserved.

When (for example) lenghts shortens, you might be tempted to think that a certain process might "gain"

by this, since a shorter distance might imply that the process runs faster.

However, time runs slower too. So a "+" on one side is balanced by a "-" on another side, so to speak.

Related to above, but more general in nature, it seems true that a SpaceTime interval between two events

always has the same "distance", no matter which frame of reference is used.

Although one often have to let that statement "sink in" for a while, hopefully you can see that such a fact

also means that "causility" is preserved.

In a way, STR is about the invariance of any spacetime interval (or the 4D distance between any two events).

Such statements needs a little more clarification ofcourse, but let's first see what we can learn

from GTR (yes, some keypoints only).

If you read articles about STR or GTR, you will often encounter the term "geodesic".

Special relativity actually plays in "flat spacetime", which we also know as the Minkowski SpaceTime.

A geodesic is the shortest path between two points, in a space where a metric is defined.

So, in a true 2 dimensional euclidean flat space (like a plane), it's a straight line.

On a 3D sphere however, it's an arc. But that's the equivalent of a straight line on such a place.

The term "geodesic" has

in STR, but I don't think it's too relevant for my discussion.

This subsection will not be too long, but not too short either. ;-)

The central idea in GTR is:

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 Newtonian sense, but a mere manifestation of the curvature of spacetime.

In order to appreciate the formulation above, we need to go through a number of points.

I do not see this subsection as "so terribly important" for the text that comes

However, it's nice if you could pick up the general idea that's presented here.

Often, vector calculus (lineair algebra), works with orthonormal (perpendicular) basis vectors.

Picture a nice clean coordinate system in Cartesian 3 dimensional space, with x,y,z axes, perpendicular to each other.

Then, points in such space can be represented by "vectors" , which can be written as the sum of basis vectors

(along the x,y,z axes) in such space.

However, sometimes we must consider curvilinear coordinate systems, such as cylindrical or spherical coordinates.

These are better equipped in many physical problems.

Sometimes, physicsts sees a difference between two types of vectors: "contravariant" and "covariant vectors".

This difference arises, or becomes clear, if the coordinate system in use, is not so simple anymore.

I mean: in that non-curved 3 dimensional space, with perpendicular x,y,z axes, as long as one chooses any

set of orthonormal (perpendicular) basis vectors, there is no difference "contravariant" and "covariant vectors".

The mathematical difference is mainly about how a vector transforms, if the set of basis vectors is changed.

Because, it should be possible to choose another basis set, since your original choice was probably one out of many

in the first place. And ofcourse, there should be nothing special of a certain set of basis vectors.

Sometimes, the "space" you want to describe with a superposition of basis vectors, is not ideal.

There might be non-perpendicular "angles" between the basis vectors, and possible they are even not normalized.

- If the vector "transforms" like the basis vectors, that is, the same (mapping) Matrix can be used, the vector

is called "covariant" (or invariant).

- If the vector "transform" contrary or opposite to the basis vectors, they are called "contravariant".

Again, in a Cartesian system, with any set of perpendicular basis vectors, there is no difference at all.

So, it can be hard "to visualize" what exactly is meant here (without math).

Yes, without math it's difficult to see what is going on here. However, I like you to remember that "covariant"

means "invariant", thus invariant means independence of a particular choice of coordinate system.

Maybe this "trick" helps: The "normal" vectors, like those common ones as a superposition from Cartesian perpendicular basis

vectors, "look" more contra-variant than co-variant. So, a position in x,y,z 3D space, looks contravariant.

Also, if you change the basis vectors, the components of the vector changes too.

It looks like nonsense talk, since in Cartesian space, with any set of perpendicular vectors, there is no difference....

Or, as another example, "tangent" vectors along a curved surface are contravariant too.

The "tangent" vectors are like "tangent-lines", and thus look rather Cartesian.

Now, to make some light in the darkness: contravariant vectors are often simply called "normal vectors",

while covariant vectors (still vector-like) are often (and better) called "linear forms".

The covariant vectors, are still vectors, but have often a mathematical "souce", like the example below:

An example of a covariant vector, would be a row vector, which is obtained from the "gradient" (differential) of

a

It expresses the local changes of Φ in the x,y,z directions, and is expressed by a row vector.

So, the ordinary "column vectors" we all know so well, are contravariant vectors.

Often, covariant vectors are expressed as "row vectors".

If you happen to know the "bra's" and "ket's" from Quantum Mechanics, then a "ket" is an ordinary

contravariant vector, and a "bra" is a covariant vector from dual space.

If you say: "what a big deal for some names...", well, the disctinction between covariant and

contravariant vectors is necessary when dealing with curved spaces. And that's where we are heading.

But I agree, we should not make a big point out of it. But, almost all articles on GTR, will mention

the terms covariant and contravariant, so that's why.

There is nothing to understand about the following. It's just a convention. How we write down

covariant and contravariant vectors is shown below. A vector is a superposition of basis vectors, and

the Σ (summation symbol) denotes that. However, in advanced articles, it is often left out,

as it it supposed be be implictly present.

Contravariant vector

Covariant vector

where

"superscript", or "subscript" too. But that's not so very important, I guess.

Note 2: Tensors.

So what is a tensor?

First, lets imagine a very large and completely empty space. It contains absolutely nothing.

No matter, no radiation, no virtual particles etc..

You can wonder if something like that could be possible in our Universe. "No", would probably be the best answer.

German has a nice term for such an experiment: "

By the way, Einstein once said:

My theory (General Theory of relativity) proves that space and time would disappear along with matter."

Suppose now, you had a very advanced small probe to your disposal, full with extremely delicate instruments,

which could detect any form of "stress", or "force"

Then, you send the probe away, on a mission in that space.... Would it detect anything?

Next, we would repeat the experiment, this time in "real" SpaceTime, which we believe is "locally curved",

since not too far away, a large Mass is present (like a star).

Again, we send the probe away, en let it take a random path in a relatively large volume of that space.

We also make sure that the probe's path includes "loops", so that we are sure we measure all possible effects

that this space may exert on our probe.

If we later on analyze the results, we may find stresses along the "tangent" (vector) directions of the curved space,

but also stresses on covariant directions, if the probe went off "it's natural" path.

It looks like, that

What's more, it seems that our probe went along a field of such variations, present at every place.

We might say that some sort of mathematical "construct" is possible, at every point, which includes the variations found.

Furthermore, all elements of this construction are defined geometrically from within that "space" itself.

So, it looks as if it's intrinsic to that space.

Well, it seems we have found a mathematical construct, which might be called a "tensor".

To see if it "really" is a tensor, we could recompute the same construction in different coordinates.

If we get back a similar construct, we probably have a tensor, according to formal definitions.

Fig. 4: illustrating simple examples of scalars, vector, and tensors.

A tensor may be viewed as a generalized mathematical object.

The "smallest" tensor, is a scalar, which is just a number. Sometimes, folks call that an "order"

(or index, or rank) "0" tensor. You do not need any index, to describe the elements, since it's only one number.

Next, comes a vector, which is an n-tuple. Sometimes, folks call that an order "1" tensor.

You need 1 index to describe (or address, or "walk" through) all a

Following that, we have an "order" "2" tensor, which looks like a Matrix. You need 2 indices, (i,j),

to describe (or address) all a

You know how a Matrix look like?

It could be a 2x2 sized one (for 2D space), with 4 numbers (for example A

Or, it could be a 3x3 sized one (for 3D space), with 9 numbers.

Or, it could be a matrix for a 6 dimensional vectorspace. Then it would have 6 columns and 6 rows, having 36 numbers.

However, in all cases, we still only need two indices i,j to describe (or address) all A

In General, we may have a tensor of "order" (or index, or rank) "n".

A "order" (or index, or rank) "3" tensor would look like a "cube", but higher than that is hard to visualize anymore.

Formally, a tensor would be described by "μ+ν" indices where μ is the number of contravariant indices,

and ν is the number of covariant indices.

Don't worry too much about contravariant and covariant. In a subsection above, I tried to explain that

sometimes to create a set of basis vectors for some "difficult" space, may not always be a clean

orthonormal (Cartesian-like) set of basisvectors. Sometimes, tangent like basis vectors are OK, and sometimes

non-tangent vectors must be used. It may be mix of such vectors.

So, to stay general, a certain rank 3 tensor might be written as:

that the term "order" is the best term to denote the number of indices.

What I think is more of interest, is this:

In a physical "every day" interpretation, we might say that every index says something about the stresses or torques

along a particular dimension, and that all components (all numbers in the tensor) produces a general overview

of all stresses or torques in a particular point in that space. So, in a certain "volume" of that space,

we have a "tensorfield" that describes all variations of those stresses or torques.

Note 3: Curved Space (Riemann, Cauchy and others).

This subsection will be very short. I just want to say that quite a few mathematicians, and physicists,

pondered on curved spaces since a long time back.

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 scientists to name them here.

For some theorems in that realm, we can go back to the years around 1850, or even earlier.

Even long before that, quite spectacular theorems were proved.

for a spherical surface, the inner pressure is always higher than the outer one. This proved correct for materials,

like a water droplet. So, for many problems with respect to surfaces, boundary conditions, spaces etc..,

they were explored, mainly by mathematicians.

I'am not sure why I mention it here. I guess that the "history of science" is one of the most facinating area's

to study.

In a later section, we will see that ideas about a "discrete space" is no fantasy at all. Up to now,

(possibly except the Planck's Length of section 1), we always have assumed a smooth SpaceTime continuum,

but in some theories, such a picture does not fit at all. Here too, the basic ideas go way, way, back.

Here, the curvature of SpaceTime was not an issue, and SpaceTime is "Minkowski-like", or, if you want, it's a flat

Eucledian (ct,x,y,z) coordinate system. Indeed, acceleration (or gravity) was not include in STR.

Now, GTR is a bit different. It's "general" since

and reveals again new fundamental insights of

What is more relevant, and will become more clear later on, is the description that

Maybe GTR can be illustrated by a sort of bulleted list of

Fig. 5: illustrating "parallel transport" and the "bending" of light.

GTR is a geometric theory of curved SpaceTime. It's an analysis of such SpaceTime, which gets curved

by any matter that is contained in it. The mathematical equations that result from the analysis,

governs the way how (other, smaller) matter should move and evolve.

More specifically, the curved SpaceTime

So, "gravity" is the result of the geometric properties of curved SpaceTime.

Or: gravity is represented by the curvature of spacetime, and not by a "classical" (Newtonian-like) force.

What will become more clear later on, is the description that

Underlying all this, or the

the " energy–momentum tensor" to the rate of "curvature of SpaceTime".

The equations mentioned above, are actually partial differential equations, which are a description

of a physical system. Next, we are supposed to find "solutions" for these equations, like for example

like Karl Schwarzschild did in constructing the "Schwarzschild metric", as one of the first solutions

of Einstein's equations.

SpaceTime is "curved" without us knowing about it.

Fig 5.1 above, tries to illustrate "parallel transport". Here you see a 2D spherical surface in 3D.

Ofcourse, it's a poor representation of 4D spacetime, but it will do for illustrating the following.

First, let's consider a common (flat) Eucledian space.

Suppose you are in one of the corners of a football field. You point a stick right in front of you,

and you keep it pointed that way.

Now, you start to walk along the border of that field. At the first corner, you turn 90 degrees to the right,

and you continue all the way, until you are back again at the point where you started to walk.

Do you notice that the stick has it's original orientation again?

That's really a propery of (flat) Euclidean space.

Next, you are at the equator of Earth. Again, you point a stick right in front of you, and keep it that way.

You walk to the North Pole. When you finally have arrived, you turn "right" and take the shortest path towards

the equator. At the equator, you walk towards the point where you have started your journey.

Do you notice that the stick this time did not retain it's original orientation?

That's actually a propery of curved space.

Now, that's only space. In SpaceTime, you

No. Even the the Cartesian-like framework (ct,x,y,z) says it's impossible: for at least "ct" has changed.

Is there a point to this? Well, any example has it's plusses and minusses, but here we have seen

an example that

"at least also" depends on the path taken between the points.

Actually, I hope that this simple example already shows you, that in curved SpaceTime, a free object does not

follow a "straight" line, in the sense that could be done in a "flat" Eucledian 3D space. (Here I mean that common

(x,y,z) space, where you can simply choose a set of orthonormal (perpendicular) basis vectors).

When people talk about curved SpaceTime, often it is said that somewhere a lot of "mass"

is present. For example, the Sun warps SpaceTime around it, causing the Earth to have a near circular

orbit around it (actually, it's an elliptic orbit).

However, when they meet a physicist, he/she might suggest to use the term "mass-energy", instead of "mass".

Ofcourse "mass" (m) and "energy" (E) are "equal" according to the Einstein's famous equation E=mc

This is an important reason to talk about "mass-energy".

But, that equation is usually interpreted as the mass and Energy in a frame of reference which is in rest.

If you now consider frame S' flying to the right, then the equation still holds for an object in S', but now

we consider it to be the "relativistic" Energy E

Or, which is the same, but now expressed in terms of "momentum" as well: "E

which is why some folks also talk about "energy-momentum".

The latter equation then, would hold in any frame of reference.

A metric is a sort of "official" naming for how you would describee the "distance" between points in any space.

Ofcourse, a mathematician would look a bit sour at above statement. But yes.., it can be defined much better...

In common (flat) 3D Euclidean space, the distance "ds" between (x

ds

which is actually not much more then applying Pythagoras. If you take the square root of ds

Or, if we have a point relative to the origin, (x

but have the index "i" in x

ds

Let the last term not surprise you. We know that the summation symbol Σ is often simply left out,

if it is obvious that the equation uses it (or physicists are lazy). See section 4.2.1.

In in an "n" dimensional flat Euclidean space, nothing changes much, except for the additional terms added, like:

ds

It's not that simple, you know, (at least for me) as I found out. I will return here quickly.

But I might just as well work on the other intro's as well....

It's existence is not doubted anymore, and even the experimental evidence is overwhelming.

But

In a "lightweight" setting we will explore the "traditional" singularity with it's

surrounding Schwartzschild radius (where curvature of SpaceTime is so steep, that even light

falls back), and some modern ideas as well, like the Firewall model, the Fuzzball model, and some others.

Black Holes are subject of many astronomical observations, and certainly the subject of nummerous

theoretical studies.

Wormholes might be related to Black Holes (in a way), but we must never forget that Wormholes

are

Indeed, almost all Top physicists will (rightfully) nod their heads, and tell us that we are on a wrong path.

However, even they "leave a certain door open", since there is at least

to explore the idea further.

Anyway, it's great stuff. As it happens, I already had an extremely modest note in place (or almost in place)

which touches on these facinating subjects. I hope you will give it a try.

So, in effect, I redirect this chapter to that note.

Hopefully, you like to try it. Then please use this link.

And, does it make sense to talk in such a way?

It's amazing really, that most physicists acknowledge, that such questions cannot be fully answered right now.

Some even speak of a "Vacuum energy crisis", due to conflicting theoretical results, and observations.

There are some observations we can make ourselves. They are not difficult, but still quite fundamental at the same time.

When looking at cosmological distances of 100 million lightyears or so, or even much larger, there is no clue

that space is fundamentally curved. Ofcourse,

the SpaceTime "close" to a star. Also, it has been found that "smaller" regions of collections of galaxies,

bend light strongly, as we can observe from Earth (or sattelites).

Sometimes, we even see more remote galaxies as mirrorred due to gravitational effects by that region.

This is often attributed to large "dark matter" collections in such region.

However, on a very large scale, SpaceTime still looks universally flat.

Since mass and energy are equivalent, then we might say that the energy density of the Vacuum must be very low,

since global SpaceTime is not curved (or not strongly curved).

This sort of observations, contribute to certain statements some physicists make, like for example that

the vacuum energy density is no more than 10

At the same time, when viewing the vacuum from theories like Quantum Field Theory (QFT), where SpaceTime

can be viewed (?) as an array of harmonic osscilators, then, when summing the energy over all modes, it turns

out that this sum

However, there are grounds to introduce a "cut-off" level (later more on that), which means the the sum will

be limited (instead of infinity), but still about 10

cosmological observations.

As might be seen from (1), it's probably very low (or "0") anyway. But really "0"?... that seems to be in conflict

with other observations.

First, as we have seen before, we have the socalled "quantum fluctuations".

It's reasonable to assume, due to observations and theory, that "virtual particles" popup, live for an extremely short time,

and disappear again.

We have, for example, the "Casimir effect" (see also chapter 2), which can be explained by those virtual particles (or fields).

Although some people are working on alternative explanations, the intrinsic fluctuations, are supported by Quantum Mechanics,

and it all seems pretty reasonable.

But the "Casimir effect" is not the only observation. Many physicists suspect that "naked" charged elementary particles

are actually not observed. Virtual particles are shielding them, sort of, leading to the phenomenon of Vacuum Polarization.

That is, on average, virtual particles repostion themselves as to weaken (sort of) the naked charge.

Another theoretical consideration is the "Unruh effect". Due to relativistic and thermodynamic considerations,

different observers in different frames of reference (one is accelerating), (might?) observe a different

background temperature/radiation in the Vacuum.

One good explanation is the notion that the "vacuum" is not the same as true "empty space".

As we know, the Vacuum is filled with quantized fields (or virtual particles), or thus Energy.

According to relativity theories, "time" and "length" are percieved differently among the observers.

If, for example, one of the Heisenberg uncertaintenty relations still hold, and here we mean "ΔE.Δt ≥ ħ/2",

then different quanta (energies) might be observed in different frames of reference.

But no article link the Unruh effect directly to Heisenberg. This is so, because actually we may not do so.

Unruh is related to acceleration, so my argument of different "Δt", in different frames, does not hold.

However, there still is a relation to the inequality relation, so that's why I used it to illustrate

the different background radiation.

The Unruh effect is, however, still subject of debate. But it seems that many physicists do not question it.

Note:

Would the Unruh effect be noticable in particle accelerators, like the LHC? As far as I have seen in the literature,

it seems that it is not. It might be so, that the acceleration of particles is still too low to notice

any effect. I'am not sure really. Please note that we are talking about the äcceleration. Ofcourse, ultimately,

the accelerated particles go very, very close to the speed of light.

And, we might also assume that the intrisic energy = "0". Would something like that be possible?

It's totally hypothetical ofcourse. But in such a Vacuum, would then the quantum fluctuations still occur?

In such a scenario, we might have a completely different "world" and completely different physics, but if

Quantum Mechanics would still hold, then we (unavoidably) still would have quantified fields (or "virtual particles").

So yes. An

Now a more realistic scenario, but it resembles (at first sight) the situation sketched above, very closely.

So, suppose we would have a

of the Vacuum itself would be "0".

Most physicists think that we are not in such a "ground state" of the Vacuum. So, there exists

some certain amount of energy per volume, in the "real" Vacuum.

The

as the

The way the Cosmological Constant was introduced into science, is mainly due to GTR from Einstein.

One of Einsteins GTR field equations, may be written as:

Einstein introduced the term Λ to counteract any possibility that the Universe might "collapse".

Mind you, the common view of the Universe in the early twenties of the former century, was that the Universe

resided in the "steady state". Indeed, at the time GTR was developed, the notion of an expanding (or collapsing) Universe was really

sort of "far fetched", and not based on any sort of decent evidence.

In the upper field equation, the curvature of SpaceTime (G

So, for the Universe as a whole, to remain static, we must not have so much mass-energy that the curvature

making the equation in "equilibrium".

Without it (g

the Universe would collapse at a certain time.

It was only later, that the "redshift" of light was discovered, meaning that galaxies that were

seemed to fly away

This is indeed not exactly a clue for a collapsing Universe, but for an expanding Universe.

Historically (say throughout the first half of the former century), three main cosmological models existed:

- the Universe is static (steady state),
- if the Universe has sufficiently mass, and it is indeed expanding, then the expansion will stop eventually,

and it will collapse again, due to gravity - the Universe is expanding but steadily at a slower rate, but there is insufficient mass for the expansion

to reverse, and slower and slower the expansion continues, and at a certain moment, the Universe will be so dilluted

that it dies out eventually.

but at a decellerating rate.

However, fact is, that after the discovery of Hubble (and others) that the more remote galaxies are, the more they seem to

recede (fly away) faster and faster. Actually, that observation does not fit nicely in those 3 models.

At a certain point, a better interpretation was deviced. It's not so that very remote galaxies go faster and faster

in terms of relative speed (as we seem to observe), but space

So, all the intermediate space between us and a very remote galaxy, really had all the time to expand

significantly, and still does so.

Indeed, in such a view, it's then reasonable that nearby galaxies do not show such a large velocity,

since the space between us and that nearby galaxy has not expanded that much.

It's not unsimilar to a common elastic string. Suppose you have an elastic string of 1m of length, unstretched.

Suppose, on the far left you imaging that our own galaxy is located. Next, you imagine that 5 cm to right, a nearby

Galaxy is located. Then, you imagine that on the far right a very remote galaxy is positioned.

Now, you strech the string to a lenght of 2m. The nearby galaxy has "moved", but it's only a little bit:

just a few cm more to the right. However, the amount of "space" between us and the remote galaxy has increased dramatically.

If light travels through a signifact "piece of expanding space", the wavelenght increases (it becomes stretched),

and that shows up by the fact that the color of that light shifts more and more to the "red".

Due to what we know of the Doppler effect, reddish light from an object, can be associated to the velocity that this

object has, flying away from us. Contrary, if the wavelenght shortens, the object moves towards us (blueshift).

It's really not much different from what we experience in normal life: if a race car speeds towards you, you hear

a highpitched noise. After it has passed, and now races away from you, the tone (frequency) drops.

Did you noticed above, that Einstein just introduced g

with the motive not to let the Universe collapse (if sufficient mass-energy or gravitation is present)?

So, Λ can be interpreted as a "repulsive term", working against a gravitational collapse.

It all started out in a rather "mundane" way. Astronomers are always keen on having reliable ways to measure

distances in the Universe. They have an array of methods, where some are more suitable for shorter distances,

while other methods are usable for large distances.

Astronomers have found that in the (relatively) "nearby" Universe (still covering hunderds of millions of lightyears up to

a couple of thoundends of millions of lightyears), the redshift of light can be reasonably accurate coupled to distance.

Having a "standard candle" for using at measurements of large distances (intergalactic, so for remote galaxies),

would be great! Ofcourse, having a source of light which is pretty "constant"in it's output, give you a good means to

associate distances with the intensity of light.

If you are interested, you may Google on

again meaning that it's intensity of the supernova, depends foremost on it's distance.

Indeed, the intrinsic total luminosity is rather constant (about a month or so), apart from it's beginning and end.

It must be said that supernova explosions are quite rare, so a good study may involve lots of telescopes around the Globe,

over months or longer.

So, as some say, it became apparent that

in the relatively nearby ranges in the Universe (up to about 1000Mpc, or about 3000 million lightyears).

After sufficient data was accumulated, interpretations seem to suggest that the luminosity of more remote supernova

were actually (and this is the crux)

Ofcourse, the actual techniques are much more involved, but typically, the results are often shown in graphs which show

the observed magnitudes plotted against the (socalled) redshift parameter z.

The plots thus aquired, seem to suggest an accelerated expansion, instead of a expansion which slowly decreases.

It's a fact that the physics of this particular supernove is well-known, and so is it's

So, given a certain distance "d

then it's

However, if the Universe expands, the distance obviously is not static, but was increased since the light was emitted

all the way up to the time we observe it.

One step further: if the Universe exposes some sort of

This scale factor "a" would then be (probably, as many assume), in the form 1/(1+z).

As an example: suppose that the redshift measured would be z=0.6. Then the scale factor would be about 0.62.

So (as many thinks it works), at the time the supernova exploded, the distance to us was about 0.62 of it's present distance.

Or,

and a nova was "r

we start to observe it, might be written as r

Since the late '90's and early 2000's, the idea of an accelerated expansion sunk in in the scientific community, and perhaps

some folks were quite surprised.

By now, a few "mechanisms" have been (theoretically) explored, and this research remains a very active field, of what exactly

drives the acceleration. The following main lines of thought can be distinquished:

- "Quintessence" as some yet unknow force or field, responsible for the current accelerated expansion.

Some argue that it may even be*time dependend*, meaning that it might have been dormant for a long time,

then, rather suddenly, picked up momentum (as some observations seem to suggest. - "Dark Energy", as the field resonsible for negative pressure on SpaceTime.

Since it's not clear what sort of field it is, it is labeled "dark". - The "cosmological constant" (which maybe synonymous with Dark Energy) could be a candidate too.

Remember that Einstein introduced this term as a "repulsive" term in his equations, to make sure his equations

did not let the Universe collapse. Again, it's still not entirely clear what exactly the "cosmological constant" is,

other than that it surely can be associated to the energy difference from a true empty Vacuum.

Many physicists vote for the cosmological constant as the repulsive force. - Socalled "Brane-World" models too, have provided different scenario's for an accelerated Universe.

We will return to these subjects in Chapter 8.

Let's now take a look at some ideas on Discrete Space and spin networks.

The principle of "Lorentz covariance" is sometimes used as an argument agaist discrete SpaceTime.

"Lorentz covariance" might be described as the requirement that there should not be a preferred frame of reference (from STR).

You might take a look at section 4.1 again, because that is essentially tied to "Lorentz covariance".

Related to this (but not the same), were (and often still are) requirements that events and equations should not

fundamentally differ if we change coordinate system, or rotate a system, or mirror a system, or replace all charges with

the opposite charge etc.. However, these "rules" or "postulates", not always have proven to be "true" under all circumstances.

But, "Lorentz covariance", which is strongly tied to Relativity theories, seemed to be viewed by a large number of physicists

as a true piler in physics.

is in conflict with discrete SpaceTime.

The following article is "just one" of them. You might take a look at it, or just read the "abstract".

It's not that I recommend reading it: it's just

with a discrete SpaceTime. This particle article uses "loop quantum gravity" as a basis for their considerations.

About Lorentz invariance in a discrete quantum setting (arxiv.org).

In general, it is viewed by most modern physicists, that "at the front" (e.g., near the cut-off scale of the Planck length),

there are ways to harmonize Lorentz covariance with discrete SpaceTime, or to simply avoid collisions.

Related to Lorentz covariance, is "diffeomorphism covariance". While Lorentz invariance is especially used when talking

about "Lorentz boosts", that is uniform moving-, or accelerating frames of reference, "diffeomorphism covariance" is more general.

A rather incomplete (and possibly wrong) way to describe "diffeomorphism covariance" is the following:

A "physical law" should not fundamentally differ when transformations or rotations takes place.

So if you would describe a system from a certain coordinate system, you may change to other coordinates, or even to a whole

other place, but your "physical law" still holds.

In other words, we should have a full coordinate-independent, and location-independent, formulation of physics.

Well actually, and possibly amazingly, such an interpretation of "diffeomorphism covariance" might be somewhat wrong.

Some formulate it as this:

This might be percieved as very strange, since, was it not that for example GTR sees gravity as the consequence

of "curved" SpaceTime??? Yes, but SpaceTime is *only* the background, and nothing fundamental, really.

So, a smaller mass may curve SpaceTime nearby, producing gravity, and a larger mass may curve SpaceTime in the same

way at a larger distance. SpaceTime itself, does not matter.

Some people like to say that physical theories should be "background independent".

In a way, you may say that still lots of physicists like "continuum physics". But, since (around) 1985 or so,

it seems that ideas of a discrete SpaceTime found ground more and more, especially with theoretical physicists.

This is not to say that way before the early 90's, many ideas already existed. But especially "loop quantum gravity"

gave it much more momentum.

But, the principle of "diffeomorphism covariance" could be seen as a serious argument against discrete SpaceTime,

since the idea of "

but difficult in discrete SpaceTime. Why? By the very definition of "discrete", there is some "black-and-white"

in SpaceTime, so "background independence" is hard to realize (even if it was only on a very tiny scale).

However, many want to pursue it further. Why? In many cases, even good-old pure Quantum Mechanics shows us discrete levels.

Just think of this: energy states of electrons (in an atom) are discrete levels. Countless other examples

can be listed, for example discrete spin states, and other quantum numbers associated with elementary particles.

It again and again boils down, that continuum theories need to be reconciled with Quantum Mechanics,

or Quantum Field Theory.

A series of very interesting "lattice-like" theories have emerged during the second half of the former century.

Also, the ideas of Roger Penrose should be mentioned too, in any note on Discrete SpaceTime.

His "spin networks" are great!

published around 1970 or so, is quite remarkable, I believe.

By the way, the ideas of Penrose contributed significantly to a number of theories, such as Loop Quantum Gravity

So, let's first take a look at some ideas of that more modern Theory: that is, "Loop Quantum Gravity".

structure of SpaceTime itself. Or is that a bad formulation?

Physicists started out to try to apply

instead of the good-old Quantum Mechanics. Folks then at first focussed especially on the usual SpaceTime metric, which is used in GTR.

But, it turned out that massive difficulties came up. In fact, the first attemps really failed.

Maybe it could have been known right from the start. In a common early conception, quantum fields were supposed to live on the background,

or SpaceTime, from the theory.

In contrast, the GTR equations like "G

before, may be considered to be background independend. The equation is valid everywhere, and thus "position" in space, and time,

are not specifically mentioned in such equation, in a natural way.

Above was not the only problem. A QFT calculation of the sum of all energies of all harmonics ( ∫ ℏ ω dω)

of a basic harmonic oscillator, would yield a tremendously high value of energy in SpaceTime.

So, a new perspective was needed. Already in the early '30s (and later again in the '60s) of the former century, some serious work

was performed on such alternative perspective. One must certainly not overlook the works of Roger Penrose. In fact, it will turn out

that later work on Quantum Loop Gravity, looks remarkably to Penroses 'spin networks'.

Then, say in the second half of the '80s, physicists more and more realized that actually two main approaches existed:

the

Ofcourse, both concepts existed way before that time, however it seems that the physicists working on Quantum Gravity

expressed the differences louder and louder.

- The covariant way, is what people were actually used to, and SpaceTime as a background is clearly visible.

Here you can see that SpaceTime geometry is the backstage on which particles and fields are manifest.

- The canonical way is like we see in GTR, and the SpaceTime background is not explicitly part of the equations.

So, it's not the explicit backstage on which particles and fields are manifest.

For loop quantum gravity, this translates to that it's basic concepts do not presuppose the existence of a given

or fixed spatial/time background metric.

One important motivation to use a canonical view, is the realisation that we might have a "integrated world", and

contrary, picturing a backstage with entities in or on it, might not be fundamental.

By the way, I just found a terrific (non-technical), but very informative article on "Loop Quantum Gravity".

I absolutely recommend to take a look at it. If you want to read it, use this link (pdf).

Nummerous chalenges exists to arrive to a quantum description of SpaceTime. For example, quantum fluctuations on the

tiniest scales (so that "+" and "-" are getting very relevant), might even break causality.

This explains why some models try to seperate Space and Time from SpaceTime, and why even "

models were designed.

Indeed, you should also realize that time is a definite part of the background. Secondly, the

is such a fundamental idea in Quantum Mechanics (and in classical theories too, like Newtonian Mechanics), that it's really

a challenge to get to models that are not directly background dependend.

So, to device a quantum description of SpaceTime, or gravity,

Also, as another thing, using the familiar "quantum way", that is creating a superposition of certain basis states of SpaceTime,

is difficult, or probably impossible.

What may rightfully surprise you, is that most initial dissertations seems to start with a

since they still using a GTR metric, but this time it's a "tetrad", which functions as a generic one.

Then, using various established theorems from mathematics, and other very involved math, the relations can then be rewritten

in a

that work in a sort of transformed phase space.

It's important to note that the fact that quantum operators describe actions/events, is a key in regarding a theory as "quantum".

One of the essentials one can fishout from various articles, is that the distance between two points becomes a quantum operator.

As it seem to appear to most physicists working in this field, the theory most essential themes are "quantum kinematics" and

"quantum constraints", which initially lead to "loops", and in later work, to "spin foam", at the Planck scale (see chapter 1).

A rather well-known interpretation of GTR is the Palatini formalism, where the "tetrad" and the "connection"

are regarded as independent dynamical variables.

The "tetrad" field (the vierbein) can be considered to denote a set of orthonormal vectors e

Sometimes, a "tetrad" is also used to denote a

transformations, is more in the line with "connections".

A "connection" has a more abstract description. Officially, one should describe it in terms of "gauge formalism"

in relation to a "fibre bundle" (field) over a manifold (subspace) "M".

However, it can be used in a more simpeler context: how can you "connect" different fieldvectors in different places?

The answer is, that it can be a "parallel transport", in the sense as we have seen before in section 4.2.2.

Or, very very generally stated: it's a transformation of some sort.

By the way, it's getting rather painfully clear, that it is absolutely difficult (or maybe impossible) to say

something useful on advanced topics of physics, without a basis of discussion of the

I am so silly sometimes! So that will be my next chapter.

In GTR, geodesic motion is a key point, whereby objects move in a way dictated by the curvature of spacetime, and masses

at different places, may move very different relative to each other. It was realized in GTR, that the "inertial frames of reference",

that is, the flat 4 dim Minkowski SpaceTime, as we know it from STR, do not exist globally in GTR.

However, in a sufficiently small volume, the curvature may be small, and SpaceTime may "look" again (a bit) like the

flat 4 dim Minkowski SpaceTime, but it

This is why in many articles, transformations are discussed, using "connections". It is so that using tetrad fields describing

a flat tangent space at every point of spacetime, then "connections" can be used to transform to "real" curved SpaceTime.

A "spin connection" (which is a bit of a specific type of connection) might be loosely described a connection needed

to cover by local rotations too.

For quite some time in developing this type of Quantum Gravity, it all still did not work out.

In 1986, Ashtekar then found, or proposed, a new sort of connection. It's a mix of "extrinsic curvature" with a "spin connection" added,

Incredably, usage of these new connections simplified the theoretical framework enormously, and ultimately a consistent set of

But we still not have arrived at fundamental "loops" or "spin foams".

Somewhat later, especially by Rovelli and Smolin, both physicists considered what would happen to the start and end values

of a field vector travelling a

Especially an old familiar "closed loop", formerly tried in field theories and QCD, proved to be usefull, namely the

It turned out to be a true "gauge invariant" solution, which is tremendously important for physicists.

Well, although the story above may be felt as hopelessly inadequate, it now might become clear that a Quantum description of Gravity works

if we base it on discrete loops.

What further is nice to know, that if we fill in known constants in the related equations, it turns out that the scale of the loops

is in the order of Planck's Length. That's pretty cool too.

Many physicists in this particular field, call the theory a

of Gravity. But many others have some questions. For example:

- How does the "classical" (continuum) larger scale SpaceTime, emerge from the Theory?
- Coupling of Matter with Gravity is still not obvious from the Theory.
- It must be noticed the Theory might say useful stuff on unifying Quantum theories and General Relativity,

but it does not fully account for elementary particles and interactions, like the strong interaction,

the elektro-weak interaction etc.. In that sense, String Theories seem to be more general. - There still is a possible problem with "time", if "time" is regarded as a SpaceTime component in the form (ct,x,y,z).

It's not easy to view time as "quantized", or "looped", and possibly even issues with repect to causality pop up.

Also, much later, he presented quite "remarkable" articles (and books) on a variety of subjects.

While several attempts on

be attributed to Penrose.

The following is just a nice article which gives you a good glimpse on Penrose's idea's on Discrete SpaceTime:

On The Nature of Quantum Geometry (Penrose, 1972)

I'am not saying you should read the entire article, although I surely recommend that. But even browsing through the first few pages

gives you a good idea on the motivation of Penrose, and his general approach.

In the remainder of this section, I will try to illuminate some of his core ideas in a few words.

Indeed, the ideas are still relevant today.

There are quite a few physicists that acknowledge that his idea's were quite original, and fundamental

for present work (like for quantum gravity and string theories).

Below you may find some bulleted points, which is in no way representative for his entire work.

I just mention this limited list, since it seems reasonable to include some remarks in a note like this one.

Penrose developed his Twistor theory during the '60s. It is a 4 dimensional complex Space

Thus it uses coordinates as complex numbers, and thus it has 4 complex dimensions. Some physicists associate twistors

also with he "4D complex Weyl spinor representation".

A single twistor is thus a point (or vector) in this twistor space. However, much more is (possibly) "encoded" in a 4 Dim complex Space,

compared to the usual Minkowski SpaceTime.

A nice example is this one: twistors with "zero twist", seem to correspond to light rays or null lines in Minkowski SpaceTime.

Amazing things happen, if mappings are constructed to and from

For example, a point in Twistor Space, may be a line in Minkowski SpaceTime.

As another example, a bundle of twistors (converging or diverging), may correspond to points in SpaceTime which get

closer to another, or just increase distance between them.

There were (and possibly still are), some interresting intersections with Quantum Gravity and String Theories.

It can (at least) be said that elements of Twistor theory were used in several Quantum Gravity models.

The "spin" of elementary particles never stops amazing physicists. A spin is found to be Up or Down, when measured

along some dimension (some axis).

Spin is like the classical orbital momentum, but in Quantum Mechanics, it's discrete and has specific values.

It's not "fully sure" as to what it is, but a simple vision could say that it "has something to do" with a rotating or twisting motion.

There exists elementary particles having half-integer spins, like 1/2, 3/2, 5/2 (fermions).

And, indeed, we have elementary particles having integer spins, like 0, 1, 2 (bosons).

Another amazing fact is Pauli's exclusion principle. No two or more fermions occupy the same position, and spin direction.

For bosons, this principle does not apply.

All in all, spin is an implicit coupling with SpaceTime, and it's still not fully understood.

Now, a few words on "Spin networks":

In several articles (like the one above), Penrose sketches multiple arguments to abandon the notion of the "continuum" of Space.

One of his main arguments is based on Quantum

Furthermore, as Penrose argues, the idea of "points" in Space is nonsensical, since you could endlessly zoom in without

ever reaching any end to such a procedure.

According to Penrose,

This does not assumes a lattice like structure, with a fixed spacing (in which the term "spacing" would be absurd anyway).

Instead, the basic building entities of Space, thus some discrete structure, simply follow combinatorial rules

in their alignment and direction in "Space".

In the last sentence, the use of the words "in Space" would ofcourse be fundamentally wrong.

What would be more correct is the phrase: their alignment and direction forms (or is) "Space".

Next, he sees as a fundamental candidate for the buiding block, the concept of "angular momentum", or "spin".

Spin units form a sort of "fishnet", while the fundamental idea is that "spin" intrinsically creates "Space", or in fact,

Now, in this theory of "spin networks", you might think that the knots in the fishnet are the "units" (so to speak)

which is "Space". As I seem to understand it, it is really

an integer multiple of spin units, which all together form "Space".

Then, he further deepens the Theory, for example, by deriving a Hamiltionian, and transformations (Poison brackets).

Maybe I return later to this subject. It's facinating stuff, but that's true also, for a large range of other topics.

So, let's go to my next section.