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.