String Theory's Graviton and Spin-2 Particle Controversy

According to the current physics dogma, the graviton is a spin-2 particle. One argument for this involves the fact that Einstein's field equations are rank-2 tensors. Rank-2 tensor corresponds with spin-2 particle? If that's the case, then a 1/2-spin electron should correspond with a 1/2-rank tensor--but there is no such tensor! Just when you've given up hope that the graviton is a spin-2 particle, another argument is thrown against the wall to see if it will stick. In the following video, Leonard Susskind presents that argument:

There is a rule that says that left-moving energy must equal right-moving energy along a closed string. Each point along the string (represented by sigma) could serve as a starting point and the end point (equal zero and 2pi); i.e., the closed string and its points are invariant. Because right-moving and left-moving energy adds to zero, and because the invariant string and its points are equivalent, the closed string must have right-moving energy and left moving energy, and those two energies must be equal. Clear as mud, right? But that's essentially the argument in favor of spin-2 particles. Here is how the graviton is represented:

The a's and b's in parenthesis are the creation operators for the x and y axis, respectively. Complex numbers are used to show angular momentum. Notice there are two angular-momentum creation terms; one for left motion and one for right motion. There are two possible states: the two angular momenta going clockwise or counter-clockwise. Now compare this setup to the photon's:

The photon has only one momentum or one spin. There is no right-left motion business. And why should there be? When angular momentum completes a cycle, the total displacement is zero. If there is a vector moving left, there is an equal, opposite vector moving right. No need for an extra spin. An extra spin is redundant. This is why the left-right-motion argument fails to justify a spin-2 particle. Nature only requires one spin to complete a cycle that adds to zero. Here is the math:

Take the integral of the entire cycle of the spin (360 degrees) and you get the equivalent of the right-left-motion business--which is zero.

Bearing all this in mind, does the graviton (assuming it exists) need to be a spin-2 particle?

Why is There An Uncertainty Principle?

According to Heisenberg, it is impossible to know a particle's momentum and position at the same time. One explanation involves hitting the particle with a high-frequency photon, so its position can be observed. The particle is so small that when the photon collides with it, the particle's momentum changes. To observe momentum, the observer must use low-frequency light to avoid changing the momentum, but then position is sacrificed.

Another reason for the uncertainty principle is the probabilistic nature of quantum mechanics. We can model this uncertainty with pairs of dice:

There are four pairs listed above. They are each indexed 11 or 22 to indicate the first pair and second pair, respectively. The idea is to roll the A11 pair and the A22 pair n number of times, take the difference, and divide by n. Doing this gives us the average difference between A11 and A22. We do the same with B11 and B22.

We then multiply the difference in A with the difference in B. That gives a number that is greater than or equal to the constant K (assuming K is really small). When n is small, the result is greater than K. When n is big, the result can be less than K--as low as zero.

The difference in A and difference in B shrink as n grows, and vice versa. It is safe to say that when we only roll one or two pairs of dice, we get a result larger than K, and we get big A and B differences; i.e., A and B are more uncertain and harder to pin down. If we roll trillions of dice (large n), then A and B are easy to pin down, and the result drops to zero. If this all seems too familiar, it should. We can replace A and B and substitute momentum (p) and position (x) to get Heisenberg's Uncertainty Principle:

Caveat: There is a difference between the dice and momentum-position: We can know either position or momentum at the quantum scale; whereas, A and B are both unpredictable when n is small.

You will find a simple way to derive Heisenberg's Uncertainty Principle here.

Debunking Bosonic String Theory's 26 Dimensions

About one hour into the above video, Leonard Susskind (one of my favorite professors) shows how string theorists mathematically derived 26 dimensions. Are you thinking what I'm thinking? If there are really 26 dimensions, then why do most string theories have fewer dimensions? This question made me watch the video with a CSI-investigator disposition.

The original goal was to get a minus-one ground state, so when a creation operator is applied, the photon will have zero mass. Imagine an x-y plane or system going down the z axis with momentum p. The frequency of the ground-state oscillator is n/2. Here is the desired equation and result:

If the result is infinity plus -1, that's OK, according to Susskind, it's just absorbed into the momentum along z and can be ignored. After some fancy math, including a Taylor expansion, the final result is ...

According to Susskind, if we drop the infinity (that lop-sided 8) we end up with -1/24. We need -1, so multiply by 24--that gives us 24 dimensions. Add the z axis and time, and that brings us to 26 dimensions. Voila!

But here's the rub: What if the result was the desired -1? How many dimensions would there be then? Let's see ... multiply by 1. That gives us one dimension. Add the z-axis and time--that brings us to two space dimensions and one time dimension!

Obviously the math and/or logic is seriously flawed.

Since it is OK to do away with infinities, we could take another approach. If you stop and think about it, we ignore infinities whenever we measure anything. How long is a meter? It's an infinite number of points. How big is a circle? It contains and infinite number of points. Everything we measure would be infinite if we didn't invent arbitrary units of measurement. We can do that here:

So instead of having 1/2 of an infinity, why not have 1/2 of a unit of ground state? And, while we're making changes, why do the frequencies of the ground state oscillators have to be a positive number when the ground state is negative? Why not be consistent? You'll note I made both sides of the above equation negative. The result is -1/2. Multiply by 2, add z and time--and voila! We live in a universe with three space and one time dimension after all!

Update: Here is another way to arrive at four dimensions instead of 26. It's based on the premise that the desired result is infinity plus -1/2. This ensures that within the infinity, there is a negative ground state.

A Simple Way to Derive the Heisenberg Uncertainty Principle

Here is a simple way to derive the Heisenberg Uncertainty Principle. It also shows why the princlple exists for momentum and position; and energy and time. Kinetic energy is the basis for the uncertainty principle. We start with that premise in step 1). (Where E=energy; m=mass; v=velocity.)

In steps 2) through 5) we use dimensional analysis to arrive at ET=PL. Energy times time equals momentum times position. (Where M=mass.)

At steps 6) and 7) we realize that ET is h-bar (Planck's constant). Since E (kinetic energy) has a 1/2 factor, we divide h-bar by 2. That brings us to 8), the Uncertainty Principle for momentum (p) and position (x).

Since ET=PL and PL = h-bar/2, it follows that ET=h-bar/2 (step 9). Step 10) is the Uncertainty Principle for energy (E) and time (t).

Why A Quantum Gravity Theory May Be Impractical

Finding that elusive graviton would be a boon to the Standard Model--but is a quantum theory of gravity practical?

According to the Wilkinson Microwave Anisotropy Probe, the ground-state energy density of the vacuum of space is E-10 joules per cubic meter.  This is essentially the energy of nothing.  Energy less than this is less than nothing.  For all intents and purposes, energies less than the vacuum don't exist.  For the graviton to exist, it should have an energy density greater than E-10.

Suppose we take two protons and place them a Bohr radius apart.  How much is the gravitational energy?  What is the gravitational energy density?

The formula used to measure gravitational energy is E = Gmm/r.  Where E is energy; G is Newton's constant (E-11); m is the proton mass (E-27) and r is the bohr radius (E-11).

E = (E-11)(E-27)(E-27)/E-11 = E-54 joules.

To determine the energy density, divide by the the volume, the cube of the Bohr radius:

E-54/E-33 = E-21 joules per cubic meter!

This is approximately 100 billion times less than the vacuum energy density!  No wonder the graviton has not been found.

Our gravitational energy formula suggests that we could reduce the radius to an infinitesimal size.  Surely we would get an astronomical amount of gravitational energy and gravitons would be as common as sand on on the beach.

There are a couple of problems with this strategy:  1.  Gmm/r can't be greater than (m^2c^4 + p^2c^2)^.5--the total energy of the protons.  2.  Most of that energy is due to the strong force and the quarks that make up the protons.  For quarks and protons to exist, they need the lion share of the available energy.

To get an energy density greater than the vacuum's there needs to be a lot more particles than just two protons.  According to the gravitational-energy formula, when you double the mass, you increase the gravitational energy four times.  The chances of finding a graviton increase exponentially as you add more mass.  Or does it?

When you add more mass, you add a haystack of particles for the graviton to hide in.  You might say that the graviton has its own uncertainty principle: the closer you get (quantum scale), the harder it is to find.  If you step back and look at the big picture (the mass of a double-star system) you can detect a very faint gravitational wave.  Such a wave has little or nothing to do with quantum physics, since you need huge masses just to get started.

These are some of the reasons why a quantum gravity theory may be impractical.

Update:  An elementary particle such as an electron at rest needs E-14 joules of energy to exist.  The gravitational energy of a proton, using the proton's classical radius, is (E-11)(E-54)/E-15 = E-50 joules.  This is not nearly enough energy to create an elementary particle such as the electron--so it appears the graviton does not exist at the quantum level, unless its energy requirement is a thousand billion trillion trillion times lower than the electron's.    

 



  

Einstein's Field Equations Simplified

Start with light speed (c) squared equal to velocity (v) squared:

Divide both sides of the equation by c^2:

We know that g is acceleration due to gravity, and it is equal to Gm/r^2. Remove an r from the denominator and we get a velocity squared:

Now we can make a substitution:

If we take the derivative of the sphere area, we get a line with a magnitude of 8(pi)r:

Multiply both sides by 8(pi)r:

We need to convert the mass (m) to energy (E), so we find mass in terms of energy:

Replace m with E/c^2 and divide both sides of the equation by r^3. Doing so gives us a number over an area or r^2. E is replaced by T (energy density). When T is increased, the r^2 on the left side must decrease. If we think of space as an imaginary sphere with radius (r), the smaller r gets, the greater the curvature. An arbitrary distance around the circumference covers a greater angle.

You might have noticed that when r gets smaller, the energy density grows larger, which in turn causes r to shrink even more. Here's where the field equation predicts a star collapsing into a black hole.

If we don't want to dwell on black holes, we need to multiply the left side by a coefficient (g), so when T increases, g increases. The variable g will also be a measure of curvature.

Convert g into the metric tensor and T into the Energy-stress tensor, and we have the equivalent of the field equations in simplified form:

But why settle for simple when you could have complicated? LOL! What about the Einstein tensor which contains the Ricci tensor and all those lovely Christoffel symbols? I show how to derive those here and here.

But what is the logic behind the left side of the field equations? Let's start with the gravitational energy minus the escape velocity energy. When those two values are equal, spacetime is considered flat and gravity is zero.

As you can see, we don't have to think of gravity in terms of spacetime. We could think of gravity as the difference between two energies. But Einstein insisted upon spacetime, we need to express energy in terms of space, or a number over r^2, so we multiply the left side by 8(pi)G/(C^4)(r^3):

Let's replace the energies with R's, then make tensors out of them and add a cosmological constant (since empty space still has energy). In the final step, those are replaced by Gij--the Einstein tensor.

Update: Below is the Lagrangian (L) derived from the field equations--further clarifying the logic behind the left side of the field equations.

Why Too Few Dimensions are as Bad as Too Many

Zero-dimensional particles and one dimensional strings present a problem: They have a finite amount of energy but take up a zero volume of space, so it’s theoretically possible to have an infinite number of these particles or strings in the smallest space you can imagine. That implies infinite energy or a lot more energy than you expect to measure.

One hydrogen atom could have an infinite amount of energy if it contains an infinite number of strings. Another hydrogen atom could contain less than an infinite amount of strings, but still have far more energy than you would expect. A third hydrogen atom has the amount of energy you expect. This is the kind of crazy universe we should have if there are things that are less than 3D. As I showed in a previous post, you have similar problems when you have more than three dimensions.

Perhaps our universe really is 3D plus time, since that arrangement avoids the absurdities mentioned above.

Is Spacetime Curved at the Planck Length?

The mass density of the universe is around 7E-22 kg/m^3. The mass density of space at the Planck length is around 5.177E96 kg/m^3! According to Einstein, a mass density of E96 should cause spacetime to fold up into a Gordian knot that even Alexander can’t undo. However, according to Newton, it’s all about the force. If Newton is right, spacetime should be flat.

Let’s take a closer look at the numbers: What does 7E-22 kg/m^3 have in common with 5.177E96 kg/m^3? They both have the same mass per meter: 5.67E27 kg/m. Or, if you prefer, the same energy per meter--which is force. In the case of spacetime, Newton is right. Whether you are looking at a big chunk of space or a tiny piece of it, the force is the same. His equation is Fg = GMm/r^2--a mass times a mass divided by a radius squared. If we assume the masses are equal and take the square root, we get mass per radius (Gm/r).

We know that g = Gm/r^2 = acceleration due to gravity. We know that velocity squared (v^2) = Gm/r, since v^2 = gr = Gm/r. For illustrative purposes, let’s set G to one. So v^2 = m/r. We also know, when it comes to spacetime, m/r is constant. If we plug it into the relativity factor, we get Ct’ = (1-m/rC^2)^.5 * Ct. Spacetime (Ct) is unaffected by m/r. When m is decreased, so is r. The mass density explodes! But m/r remains constant and does not contract, curve or warp spacetime (Ct).

We can now say with confidence, that on the Planck scale, spacetime is as flat, or has the expected value of being flat, as it is on very large scales.

Re: Expected Value--Here's Another Take:

What will we find at the Planck scale? Will we find something neat and tidy like a string or point particle? Chances are we will find chaos. To make sense of chaos, we could calculate its expectation value, and, for our convenience, treat that value as a string or point particle. This done in astrophysics. When we do calculations there we treat stars and planets as point particles--not because they are point particles. It is just easier math when we don't have to worry about their actual dimensions.

Strings and point particles are probably like purple pixels. Have you seen a purple pixel? Probably not. They don't exist, but we can imagine the purple on our computer screens to be made up of tiny little purple pixels. In reality purple is made up of a hodge podge of red, blue and green pixels. When we add up this chaotic mess and divide by the total (n), we get the expected value which is our imaginary purple pixel. If it is convenient, we can use it to simplify any math operations that involve the color purple.

Is Quantum Entanglement Really Spooky Action at a Distance?

You may have heard that you can take two entangled particles and put them on opposite ends of the universe, and when you change the state of one, the other particle also changes its state instantaneously! Einstein called this spooky action at a distance.

It does seem weird when you consider that nothing is supposed to exceed light speed. How did the first particle get the state-change message to the second particle so fast? This is a question that has spread through the physics community at greater-than-light speed. As far as I can tell, it is based on an assumption that particle one must send information to particle two before particle two can change its state.

We could start with a different assumption. If you take a good look around you, you will find that entanglement is more commonplace and not so strange after all. Take marriage, for example. What could be more entangled than marriage?

Imagine this married couple named Bob and Stella. Bob is an astronaut, so he gets in his flying saucer, flies it straight out of the hanger at Area 51, and whisks out of earth’s atmosphere to a remote exoplanet a gazillion light years away.

When Bob arrives at the exoplanet, he changes his state: he dies. His saucer crash lands and blows asunder with Hollywood pyrotechnics and special effects. Bob was alive, but now he’s dead. Don’t feel bad for him; he’s only make-believe. The important point to remember here is his state has changed. Stella’s state has also changed. It changed the instant Bob died. The moment Bob died, she became a widow. She didn’t have to wait for a signal from the distant exoplanet to reach her before she became a widow. She may have no idea that Bob is dead, and won’t be notified any time soon, but nevertheless, her state has changed from happily-married wife to widow--and it happened instantaneously!

That’s entanglement.

Another Take on Quantum Entanglement

Einstein described quantum entanglement as "spooky action at a distance." Ironically, his theory of relativity explains what is happening.

Imagine two entangled particles separated by a large distance. Let's say it takes a photon a billion years to go from one particle to the other. But somehow, the particles can send information to each other instantaneously. Obviously the particles' information transmission is breaking the light-speed limit. Or is it?

Each particle has mass, energy and its own clock. The space between the particles is a vacuum. It has virtually no mass, energy, and it too has its own clock. Its clock however ticks far faster than each particle's clock. From its point of view, it sends the information along at a speed less than or equal to light speed.

From each particle's point of view, the information reaches its destination instantaneously. We also see it this way, since we are made up of similar particles with clocks that tick far far slower than spacetime's clock.

Are the Laws of Physics Unchanged Under Time Reversal?

The CPT Theorem says if you replace matter with anti-matter, reverse charge, parity, and time, you will have a universe that is a mirror image of our own. That seems reasonable on the face of it. But then it goes on to say that the laws of physics will be the same when time is reversed????

Let’s pretend you are bored surfing the internet and you decide to go outside and watch paratroopers jump out of an airplane. You are standing on solid ground. Gravity is holding you firm to the earth. It has an attractive force. You look up. You see a guy jump out of the plane. He falls, his chute opens and you watch him glide to the ground. Gravity works the same way for him as it does for you: it’s an attractive force pulling him to earth.

Now, somebody flips the switch, and time reverses. You are still standing on firm, solid ground. Gravity is still an attractive force for you. But the paratrooper guy? Well he’s now falling up! Gravity is a repulsive force for him. Gravity is no longer consistent; it has changed!

You suddenly wake up and realize you were just having a bad dream. You are in your bedroom, time is moving forward and all is well with the laws of physics. You decide to play with your magnet. There are some iron filings stuck to it and there is another iron filing flying toward it. Electromagnetism seems to work as you expect. Then someone flips the switch. Time reverses. The iron filings that were stuck to the magnet stay stuck. So far, so good--but the iron filing that was flying toward your magnet is now flying away! Electromagnetism is messed up! You can’t cure it by waking up this time. The only cure is forward time.

Conclusion: It is no accident the arrow of time goes forward. The laws of physics depend on it.

How to Travel Back in Time

You might be aware that relativity makes it possible to travel forward in time, but going back in time is thought to be impossible due to the light-speed limitation. It has been suggested that to go back in time, we must go faster than the speed of light.

‘t = (1-(v^2/c^2))^.5 * t (1

Equation 1 assumes velocity squared (v^2) is less than or equal to the speed of light squared (c^2). The best we can do here is slow our time (‘t) by flying in our spaceship at a very fast sub-light speed. When we arrive back on earth, earth time (t) will be years ahead of our time (t’). Our grandchildren will be older than we are. The price of a stick of gum will be $1,000.98. We’ve seen the future, but now we want to go back. But not back to the present. A little before that, so we can bet on winning horses and stock investments. How do we do it? We know that gravitational acceleration (g) = Gm/r^2 (G=gravitational constant, m=mass, r=radius), and that v^2 = Gm/r. We can make the following substitution:

‘t = (1-(Gm/rc^2))^.5 * t (2

Equation 2 shows that mass (m) can also slow time (‘t) and take us into the future. But as far as we know, mass (m) has no upper limit like velocity. that means the mass equivalent of velocity (Gm/r) can exceed light speed. We can now travel back in time. All we have to do is fly our spaceship near a massive object that exceeds C^2. You might ask, “If that really works, then where are all the visitors from the future?” Good question! There are a few theories:

1. The visitors from the future are the UFOs. They keep a low profile and avoid contact for fear they could seriously alter the future. If you visit a time before your birth, your mother might fall in love with you instead of your father. You would then vanish because you would not be born. So you don’t make contact. You stay in your flying saucer and do a couple of photo-ops, but no more than that.

2. The visitors from the future disguise themselves as homeless war veterans. No one pays much attention to them, so they don’t alter the future significantly. They are watching us, though. That’s why they stand or sit at crowded street corners and watch us go by. They don’t need to take notes regarding our activities. They have a chip implanted in their brains that record everything they see. When they go back to the future, they upload the data to Facebook.

3. They flew too close to a black hole and were crushed before they could travel back in time.

4. There are no visitors from the future.

Theory number 4 is true if Gm/r has an upper limit of C^2 like V^2 does. That means a black hole’s mass per radius can never be infinite, and, if someone claims it can be, ask, “Where are the visitors from the future?”

How Uncertainty is the Root of All Certainty

If you roll a pair of dice, you never know what you will get. You might get anything from snake eyes to twelve. But what happens when you roll a trillion pairs of dice all at once? Strangely enough, you get seven-trillion or a number very close to that figure when you add up all those dice. The uncertainty has become a certainty.

When you calculate the average dice throw, the number you always get is seven. Seven is the mean number, and two and twelve are the extreme numbers. Even if you were to eliminate all the numbers between two and twelve, you would still have an average of seven.

(2 + 12)/2=7

There is an old theorem in quantum mechanics (Ehrenfest theorem) that says that all our laws of physics that give us certain results are rooted in uncertainty. The “certain result” is called the expectation value. You calculate the expectation value by simply adding up and taking the average of all the uncertain results.

At small scales, matter behaves in very strange and unexpected ways, but at large scales, matter behaves as we expect: it follows the laws of Newtonian physics. It behaves much like a pair of dice on the quantum scale, and behaves like a trillion pairs of dice on our old, familiar scale.

Yet Another Dark Matter Hypothesis

What exactly is dark matter? How about an evaporated black hole?

A radius of a black hole is typically 2Gm/c^2 or less. (G=Newton’s constant; m=black hole mass; c=light speed.) When the radius (r) becomes Gm/c^2, the black hole’s energy is 100% gravity; as the radius shrinks, so does the mass, and the mass density increases, so does the gravity. Theoretically, the black hole could reduce to a point particle.

But here’s the question: When it becomes a point particle, is it something or nothing? If it’s something, gravity is infinite. If it’s nothing, gravity is zero. We won’t know which it is until we measure it.

Now imagine a space filled with a mixture of particles that are basically dead or dying black holes. Some might have infinite gravity; some might have zero gravity; some might have finite gravity. If we calculate the average of all that gravity we get a finite amount of gravity that corresponds to nearly empty space. What little matter there is does not emit radiation that we can detect, since this matter is 100% gravity. It’s strong, weak and EM forces were spent long ago during the evaporation process.

It is not certain that dark matter exists, but there seems to be more gravity than the current inventory of matter can explain. Here's the mathematics pertaining to dark matter:

If we start with Einstein's and Newton's equations, we can derive 3 and 4--the amount of energy devoted to gravity out of a total energy of mc^2.

Ordinary matter is represented at 3. Only a small part of its total energy is devoted to gravity. the majority of its energy corresponds with the electromagnetic (EM), weak and strong forces. We can detect ordinary matter. It emits EM radiation (light, radio waves, etc.).

Dark matter, on the other hand, has virtually all its energy devoted to gravity (see 4). It emits no EM radiation so it is undetectable.

The big bang singularity was possibly also highly concentrated matter with all its energy devoted to gravity. The other forces evolved when the universe expanded. The greater the radius of the matter and space-time, the less mass density there is, and so gravity becomes weaker and the other forces emerge.

Is Time Real or a Man-made Construct?

The measure of time is definitely a man-made construct. The sunrise and sunset, the phases of the moon were among our first units of time. But why did humans bother measuring something that does not exist? Why would they make up time and then measure it? Let’s assume our ancestors weren’t morons and at least believed they were measuring something real.

So is time real or some figment of the human imagination?

Here is one way to convince yourself that time is indeed real: Ask yourself, if time is not real, how long can a state of no-time exist? For a state of no-time to exist for any period of “time” there must be time! And thus it is impossible for a state of no-time to exist.

For anything to exist, there must be time. Existence implies time. Without time, nothing can exist for any period of time. Therefore, without time, nothing would exist.

If time is really just a human construct and not a reality, then it should not be a problem for us to do away with all our measurements of time. After all, we can live without the measurements of the dragons living in our basements because said dragons don’t really exist. They are a great example of a purely man-made construct made under the influence of those “special” brownies containing that “special” ingredient.

If we can measure velocity, acceleration, energy and momentum without time, then I would have to agree that time is just a “man-made construct.” However, those who preach, “Time isn’t real!” fail to provide an alternative model of reality that does not include time.

How would history unfold? How would events occur? Time makes history and events possible. What caused the big bang? Time! Imagine a singularity without time or space. How long can such a state last? Without time, such a state can’t last at all, so time had to begin and set off the big bang.

If you are one of those who still insists that time isn’t real, just a man-made construct, then I have a question for you: Is distance real or just a man-made construct?

That Mysterious Thing Called Gravity

Imagine you are jumping off a high-rise. From your perspective, you are falling straight down to the street below. Einstein, however, would argue that it's relative: the street could be falling to you while you stay suspended in space. If this is the case, then your mass would not matter. You could weigh a ton or be massless and the street would still accelerate toward you at 9.8m/s^2. He would also argue that your spacetime is curved. But you are falling in a straight line toward the street or vice versa. So what is curved spacetime?

Look at the left diagram. There's a circle that represents earth and there's an arrow pointing down. So where's the curve? You are accelerating. That means your change in distance per change in time is changing. If you plot on a graph the total distance you fell for each time period, you end up with a curve. If you know calculus then you know that acceleration is the double derivative of distance, and, when you calculate the double integral of acceleration, you get the curved graph you see at the far left.

The middle diagram shows how a big planet (A) has less spacetime curvature along distance (ds) than planet B. Yet, planet A has more gravity due to its superior mass density. Here is where Einstein's theory of general relativity seems to break down. Data from NASA and some number crunching confirm this lack of correlation between curved spacetime and gravity:

Are String Theory's Extra Dimensions Real?

Why do positrons and electrons have opposite charges? One explanation involves an extra dimension: the positron moves in one direction down that dimension and the electron moves in the opposite direction. The result is opposite charge. But you might ask, what prevents the electron and the positron from changing direction? Do positrons become electrons and vice versa? If they do, that would be a sign that the extra dimension exists. If not ... oh well.

If extra dimensions do exist, here is a scenario that should happen:

You have three hydrogen atoms. One has an infinite mass; the second one has the mass you expect; the third one has a mass of five trillion kilo-tons. How is that possible?

If you look carefully at the dimensions we are familiar with, you will notice a pattern. There are an infinite number of points along a line; an infinite number of lines along a plane; an infinite number of planes inside a cube. In short, a higher dimension can accommodate an infinite number of lower-dimensional objects. If this pattern is consistent, then a 4D space could accommodate an infinite number of 3D cubes. We can check this:

A cube is measured in cubic meters or m^3. A 4D hyper-cube is measured with m^4. How much 4D room does an m^3 cube take up? x * y * z * 0 = 0 m^4.

As you can see, the cube has magnitude along x, y, and z, but zero magnitude along the 4D axis. As a result, it takes up zero m^4 space. And, it can be any size! So an m^4 space can hold an infinite amount of 3D space or matter, or energy.

Now imagine there are extra dimensions in a tiny region of, say, a hydrogen atom. Particles are free to enter and exit this region. Once inside, the particles encounter an infinite 3D universe. So potentially, they could gather there and cause the hydrogen atom to have significantly increased mass and energy.

Ironically, the m^4 space could be zero and still hold and infinite amount of of 3D space, matter and energy, since anything 3D takes up zero m^4 space. So if extra dimensions exist, it does not matter how short or curled up they are. They will accommodate infinite 3D.

This presents a big problem for physics, since you could not count on two identical items having the same mass or energy. The fact that we can count on two identical items to be identical, is a sign that extra dimensions are highly improbable. More dimensions would destroy energy conservation, since up to an infinite amount of energy could leave or enter our universe at any time.

Are there other ways we can test whether extra dimensions exist?

Certainly! By definition, space dimensions are vectors that are orthogonal to each other; they are right angles to each other. So far we humans have discovered three: i, j, and k.

Suppose we think there might be a fourth dimension. Let’s call it “a.” We can test this dimension with cross products. We know that iXj = k; jXk = i; and kXi = j.

So the question becomes what is aXi? The answer has to be either j or k. If it’s j, then aXi = kXi, so “a” could be parallel to k. If it is not orthogonal to k, it is not really a fourth space dimension. If the answer is k, then “a” could be parallel to j, and our conclusion about “a” remains. aXj and aXk can yield more parallel vectors rather than orthogonal ones. It appears the cross product test falsifies the notion of a fourth spacial dimension or additional dimensions.

But suppose we change the rules. We decide, for example, that aXi = j or k. This is possible if the four dimensions are perpendicular to each other. However, physics would be less certain. We don’t know whether we will get j or k. Physics becomes even more uncertain when you add additional dimensions. If you have nine space dimensions, then aXi could equal any of the other seven vectors. Adding more dimensions to our physics does solve certain problems, but at the expense of creating new absurdities and uncertainties.

Some theoretical physicists want to discover extra dimensions so badly they can taste them--but only in their dreams. To date, there is no empirical evidence of extra spacial dimensions.

Deriving Schrodinger's Equation

Schrodinger's time-independent and time-dependent equation can be derived this way:

How to Derive Christoffel Symbols and the Covariant Derivative

Here is my new and improved derivation of Christoffel symbols and the covariant derivative. We begin with the metric. Let's convert the rank-one tensors (xixj) to x^2 and pull it out of the radical:

Next, let's take the ordinary derivative, using the product rule and chain rule of calculus:

In the last equation above, we divided both sides of the equation by (gij)^.5. Below we use identities and substitutions to put the equation into a covariant derivative format, which includes the Christoffel symbol:

Finally, we use a similar process to derive the covariant derivative and Christoffel symbol for a contra-variant metric tensor and co-variant rank-one tensor. These tensors are the inverse of the tensors we worked with above (co-variant metric tensor and contra-variant rank-one tensor).

How to derive the Riemann, Ricci, and Einstein Tensors

Here is one way to derive the Riemann, Ricci and Einstein tensors: