Quantum Tunneling Out of the Black Hole

A particle is trapped inside a black hole. What is the probability it will escape via quantum tunneling? To figure this out, we begin by defining the variables involved:

The diagram below shows two regions: I and O. Region I is inside the event horizon, within the Schwarzshild radius (rs). Region O is outside the event horizon. The question we ask is what is the probability the particle (red dot) will reach region O? And, we could also ask what is the probability the particle will fail to reach O and remain trapped in region I? These probabilities, when added together should equal 1.

Region I covers a distance (r) from zero to rs. Region O is from rs to infinity. These will be the boundaries we will be using in the integrals below.

To find a probability in quantum mechanics we are told to square the wave-function amplitude. To see how this works, let's consider finding the probability without squaring the wave function. (See equations 1 and 2 below.)

We can model the wave function using right triangles. This is appropriate, since sine and cosine represent waves.

At the second triangle above we substitute some dummy wave functions for demonstration purposes. The trigonometric proof below shows why it is important to square a wave function to get a probability:

At equations 3 and 4 we didn't square the wave functions--they rarely add up to one, so they can't be probabilities. However once they are squared, they add up to one and could be probabilities (see equations 5 through 9). You may have noticed the wave functions have negative exponents. This feature prevents an exponential blow up to infinity.

Now, to get the right values for the probabilities we also need to include a normalization factor (A). At equations 10 to 15 below, we calculate the value of A and see why we need it. When we take the sum of all probabilities, from zero to infinity, we want the grand total to be one.

Equation 16 below is our new-and-improved wave function. Equation 17 is the one we use to calculate the probability densities of regions I and O. Equations 18 through 22 yield our desired result: equation 23, the probability density of region O; i.e., the probability that the particle will successfully escape the black hole.

Equation 24 should be the probability density for the particle's failure to escape. Let's check this:

Equation 28 confirms and matches 24. Below we use Schrodinger's equation to find the value of the wave number k:

Equation 34 shows that k is a function of V--the black hole's potential. The bigger V is, the bigger k is, the smaller the wave function and the probability that the particle will escape.

For more on the topic of quantum tunneling and QM, I highly recommend Robert Eagle's (aka: DrPhysicsA) video series:

The New Relativity Particle Wave Function

General Relativity and quantum mechanics don't mix--so we are told by an echo chamber that has lasted a century. The Schrodinger time-dependent and time-independent equations don't fit with the 4D tensor format of Einstein's field equations. I decided to take on the challenge.

The root of the problem is the wave function (represented by the Greek letter psi). It depends on three space dimensions multiplied by a factor of k (momentum[p]/h-bar) and one time dimension multiplied by frequency (f).

To be compatible with relativity, we need to express the wave function in four spacial dimensions, where light speed times time (ct) makes up the fourth dimension. We multiply that by k4 to make it equivalent to frequency times time (ft).

Below are the mathematical steps needed to make an incompatible wave function into a compatible one. The variable A can be any coefficient for the exponent (exp()).

Notice the variable k has become four variables (k1, k2, k3, k4). The logic behind this will become clear when we test the new wave function later. For now, more variables offer more flexibility. They can all be equal to k or their values can vary depending on the situation.

We would very much like the the new wave function to do all the tricks the old one can do and then some. Let's do the time-dependent first-derivative test and see if the new wave function yields the same result as the old one. We set the variable A to 1.

Looks like a winner! Let's try the time-independent second-derivative test for the x-axis:

We do indeed get the same result for both wave functions. The only difference is the syntax. k becomes k1. Variable p^2 (momentum) becomes p1p1. It should be obvious that what works for the x-axis also works for y and z. So we can use this new wave function when working with Schrodinger's equation.

Now let's convert variables x, y, z, and ct to x1, x2, x3, x4. Let's see how well the new wave function works with a quantized version of the field equations. Uij is potential energy; h-bar is Planck's constant; m is mass; G is Newton's constant; c is light speed; V is volume; N is the large number of particles needed to make a little gravity; gijGij are scalar coefficients for each element of the rank-2 tensor created by the double partial derivative of the wave function psi along xi, xj.

With a little differential calculus and a pinch of algebra we can derive the field equations:

Notice how the wave function's complex conjugate is used to get the expectation value and eliminate the wave function psi.

Now I will show you the advantage of having more than one k variable. If we take double-partial derivatives of psi along k1, k2, k3 and k4, we can derive the spacetime metric and Lorentz factor. We can also derive yet again the field equations. You can see the details in my blog post entitled "Einstein's Field Equations Simplified."

OK, so we can derive a bunch of stuff with this new wave function. What's the big deal? Notice how we've unified General Relativity with quantum mechanics without invoking strings, branes or extra dimensions. Heck, we didn't even invoke the graviton. These things may or may not exist, but if they are never discovered we have a way to work with quantum gravity.

Since we don't need strings or extra dimensions to do this work, we have far less to prove than string theorists. We can make it work with established, empirically verified physics--and they can't. That puts us way ahead of the curve.