Why Gravity is Weak

Perhaps you are familiar with some of the more exotic and far-out theories that attempt to explain why gravity is so weak compared to, say, electromagnetism. One of my favorites involves gravitons hiding in extra dimensions. The logic is as follows: if gravitons weren't hiding in extra dimensions, gravity would be much stronger. It's analogous to someone screaming at the top of his lungs. The intensity of the sound is very strong until you hide that person away in a broom closet and lock the door.

As much as I like this theory, it has a problem that can't be hidden in a closet or extra dimension: there is no empirical evidence of extra dimensions. That begs the question: is it possible to explain gravity's weakness without using extra dimensions? Yes! When doing physics, I prefer taking an Occam's razor approach and using concepts and ideas that are testable and/or have been tested and confirmed. The end result is not as exciting as extra dimensions--but more likely to be true.

Let's begin with Einstein's field equations:

To simplify matters we equate the left side of equation 1 to K:

In equation 3 above, we have energy density (T44) multiplied by a constant that consists of G and c. Why do we need such a constant? Why can't we get by using equations 4 and 5 below?

The constant in question is a very small number and substantially reduces the value of K, the spacetime curvature. Perhaps equations 6 through 8 can shed some light on the subject:

Imagine energy (E) interacting with a cubic meter of spacetime. Equation 6 shows that spacetime energy density (Ts) is a tiny number. Since curved spacetime causes gravity, and since spactime's energy is so weak, is it any wonder that gravity is weak? Let's see if the math agrees. Equations 7 and 8 are the average radius of a nucleus and the nucleus volume respectively. Why are these numbers important? Imagine energy (E) added to a cubic meter of spacetime. To see how that energy interacts with spacetime, it's easier to take all spacetime's energy and reduce it to a particle. We do the same with the added energy. So we have a particle interacting with another particle within a cubic meter. Spacetime's total energy is equivalent to a proton's. We can imagine it having an approximate radius of e-15 meters and a volume of e-45 cubic meters. Our energy particle will have the same dimensions.

To provide a simple visual demonstration of the interaction between matter and spacetime, let's pretend that the volume of our spacetime proton is .25 cubic meters and that the energy particle we add has a .25 probability of interacting with it.

Because the probability of interaction is .25, the average energy ends up being .25 the total energy. To get the proper value for K, we would need to multiply E/m^3 by a constant of .25. So K is weaker than E/m^3. To illustrate the point further, let's distribute the energy evenly throughout the cubic meter. This time it will be a field of isotropic energy interacting with our spacetime proton:

At any given time, only 25% of the energy field interacts with the spacetime proton. The remaining 75% contributes nothing to the value of K. To drive this point completely home, let's cut up the spacetime proton and evenly distribute it within the cubic meter:

As you can see, it matters not how we distribute the matter energy and spacetime energy. In the case above, only 25% of the total matter energy interacts with the spacetime energy. Our constant of .25 remains constant. Now, let's stop pretending and let's replace the .25 constant with the volume of a nucleus, which is approximately e-45 m^3.

If e-45 is the right constant, then that means only e-45 of the energy contained in T44 contributes to K. Let's check it:

It looks as though it's in the ballpark of the actual constant used in Einstein's field equations. So it's highly plausible that gravity is weak due to energy interacting with very weak spacetime energy.

Unifying Spacetime and the Forces

Let's start with a simple idea and expand on it. Imagine you have a total energy (E) and you subtract from that the energy of the strong, weak and electromagnetic forces. What do you have left? Gravitational energy.

Now we need something that represents the total energy. How about Schrodinger's Hamiltonian?

It's a good idea to express Schrodinger's Hamiltonian in terms of light speed squared (C^2). Why? Because light speed squared connects all the fundamental interactions (forces)--so we will set them all equal to C^2.

If we divide the Hamiltonian by mass (m) we set it equal to C^2. (C^2 = E/m.) Next, let's multiply each C^2 by time squared (t^2). Doing so gives us the spacetime metric. To make the terms equal, we should probably multiply each one by a coefficient K.

As you can see we've unified the forces and spacetime. What better way to spend a Sunday afternoon? The last equation, which is the spacetime metric, is just another way of saying, "Gravitational energy is equal to the total energy minus the other forces' energy."

In case you are curious, here are the variables: s is spacetime; z is the proton number; epsilon is the permittivity of free space; h-bar is Planck's constant; c is light speed; t is time; gij is the metric tensor. E is electricity; B is magnetism; I is current; p is charge density; Gn is Newton's constant; Gij is Einstein's tensor; Tij is the energy-stress tensor; Mh is the Higgs mass; g is the magnitude scaling constant; e is the natural exponent; m is the Yukawa particle mass; r is the particle radial distance; Fw is the weak force.

And let's not forget the equation for the Higgs field energy. The Greek letter there is the complex scalar field. H is its Hamiltonian (energy). The next term is kinetic energy and the last two terms are potential energy.

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.