Is time proportional to the density of space in the Universe?

Answer 1

There is a relationship between time and mass: both are affected by the Lorentz Transform. And because mass is but one factor in density, there is a relationship between time and density and through that transform.

t = t0/sqrt(1 - v^2/c^2) and m = m0/sqrt(1 - v^2/c^2); where t0 and m0 are the at-rest time and mass, and t and m are the effective time and mass when the reference frame for t0 and m0 is going at a velocity v relative to an observer outside that reference frame. c = speed of light ~ 300,000 kps in a vacuum.

For example, if v = .7c, t ~ (1/.7)t0 ~ 1.4 t0 and m ~ 1.4 m0 (you can check the math for precise answers, but I'm approximating here). That is, both effective time and mass seem to increase by 40% as far as an outside observer is concerned. As time and mass are relative (which is why we call the theory, the theory of relativity) an outside observer would see the mass 1.4 times what it was on the ground and the time on the moving frame slow down to about 70% speed. For example, 1.4 years would pass on Earth, while only 1.0 year would pass within the moving frame.

As an example, assume the frame of reference is a moving spaceship going at v = .7c, 70% the speed of light. To us standiing on Earth, the mass of that spaceship would increase by 40% and the rate of time passing on board would be about 70% what we were experiencing. (By the way, all this has been repeatedly proved with quanta accelerated in super colliders.)

But you asked about density. Density is mass/volume = rho = m/V(x,y,z); where m = the effective mass of the spaceship and V(x,y,z) is the volume of the spaceship in x,y,z coordinates outside the ship. (In other words, volume is measured by how much space it takes up as an outside observer sees it.) Assume the spaceship is traveling in the x direction as we see it from Earth.

Turns out, x is also affected by the Lorentz Transform = L(v) = sqrt(1 - v^2/c^2); but in this case x = x0 L(v) = x0 sqrt(1 - v^2/c^2); that is, the transform is in the numerator, not the denominator. Thus, x = sqrt(1 - .7^2)x0 ~ .7 x0 when v = .7c Note, the y and z directions are not affected by v...only x, the direction the spaceship is going, is transformed. Thus, y = y0 and z = z0 no matter how fast the ship is going.

So what happens to volume (V)? It shrinks as v --> c and as seen from outside the ship. Inside, time, mass, and volume seem normal (i.e., at rest...t0, m0, and V0 = V(x0,y0,z0). But outside, that's a different story. We are seeing the length of that spaceship get shorter, 70% what it was before launch time on Earth.

Because V = x X y X z more or less in general, the volume of that spaceship (from our point of view) is getting smaller at the same time its mass is getting bigger. In other words, its density (of that chunk of our space the ship occupies) is getting denser. Since rho = m/V = m0/L(v)/V(x0 L(v), y0, z0) = m0/[V(x0 y0 z0)L(v)^2] = m0/[V0 (1 - v^2/c^2)]. Because the sqrt disappears from squaring L(v), we find that when v = .7c, L(v = .7c)^2 = 1 - (.7)^2 ~ .5. Thus, for our spaceship, rho(v = .7c) = m0/.5V0 = 2 rho0; where rho0 is the rest density when V0 = V(x0 y0 z0) = rest volume. In other words, while mass is increased by 40% (1.4 m0), the ship's density is doubling because its volume is decreasing as v --> .7c.

So the answer is, no, time is not proportional to space density, but, through the Lorentz Transform, both are related to the velocity of the frame relative to the speed of light. In other words, both time and density are affected by how fast something is going; so both are proportional to a common factor...the velocity relative to the speed of light.

Answer 2

"Space" by definition is the total absence of any material substance, i.e., vacuum. Therefore "space" has no density.

Light moving through an absolute vacuum (..space..) moves at 299,792,458 meters/second. However, even in the voids between galaxies there are one or two molecules per cubic centimeter. Light moving such regions might 'only' move at 299,792,457.98 meters/second. Therefore it would take light more time to get from point to point, so in that regard time is proportional to the density of "space."