The Quantum Mechanics of Gravitional Waves

Today we are going to work out the quantum mechanics of gravitational waves using my relativity wave function. If you have no clue what I'm talking about, check out my post entitled "The New Relativity Particle Wave Function." The following video provides some basic information on g-waves:

Of course there are always naysayers. Here is a skeptic's video:

I didn't find the magnet demonstration very convincing. Magnets do a great job accelerating iron filings, but not wood chips. If the guy in the above video got those magnets to accelerate something non-metallic, I'd be impressed, but all he did was show magnetic field interactions under a bright light. Thus it seems like a safe bet to build a theory of quantum gravity based on the gravitational wave discoveries.

Below are some gravitational-wave equations we can work with. The first one is the chirp mass (Mc). It depends on two very big masses: m1 and m2. They could be a double-star system or a couple of black holes.

Next we have the scaling amplitude (q)--the size of the wave's stretching and squeezing (here is a link to an animation). Normally, the variable h is used, but later we are going to use h for Planck's constant along with h-bar. G is Newton's constant; c is light speed; distance is d; frequency is f, and there's pi.

Then there is the chirp (dot f). (For a demonstration of the chirp, see the video below .)

Next, we have the gravity-wave phase angle represented by the Greek letter phi. As you can see it depends on time (t). Below it is the chirp waveform amplitude q(t).

We can take q(t) and convert it into a wave-function expression:

As you can see, if we use Euler's identity and add the exponent to its complex conjugate, multiply by 1/2q, we get q(t). We now have a quantum expression we can use.

Since phi is time-dependent, and since the goal is to equate it with the relativity wave function, it is imperative that we make the relativity wave function time-dependent as well:

Note how position variables x, y, z were converted to time variables t1, t2, t3. We add these with t4 and multiply by coefficient a to get the observed time t. The four time dimensions may or may not add up to t which is why we multiply by a. We also multiply kct by A, B and C to make them equal to each expression in the phi function. We add those to get the final result: Dkct=phi(t). We use these equated expressions in the equated wave functions and their complex conjugates below:

Now we have the tools we need to calculate the gravitational waves' energy (E), the entropy (S), the temperature (T), the momentum (p), the velocity (c) and the acceleration due to gravity (g). Let's plug these wave functions into Schrodinger's time-dependant derivative expression and see what happens (but first we need to multiply both sides of the equation above by 2/q and subtract the complex conjugates):

Multiply both sides by a complex conjugate to get the expectation value:

At last! We get the energy (E). With a little algebra we can determine the entropy (S), the temperature (T), the momentum (p). We also get the velocity, which happens to be the speed of light (c). If we divide the velocity by t, we get an acceleration (g).

So there you have it!--a theory of quantum gravity based on something scientists claim they have observed and measured: gravitational waves.

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.    

 



  

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: