The radius of the universe and time

Question

Ok - fair warning - Non-physicist asking dumb-assed questions here again. I've been reading a lot of Einstein, Feynmann, Ferris etc. I'm just loving this stuff.

But I suddenly found myself looking at the 4-D sphere concept of our expanding universe, as well as the distortion of spacetime caused by objects with mass and so on, and suddenly thought - this 4th spacial dimension (that we cannot comprehend) through which our universe is expanding - is this actually the dimension of time?

Is the time dilation experienced closer to bodies with mass not a literal curving of the skin of our 4-D sphere in toward its center? So at that point in spacetime the radius is increasing more slowly compared to the radius further from the mass?

I'm afraid my brain just keeps failing when I try to incorporate other concepts from GR into this view - such as time dilation with increasing speed. However the shortening of length of an object in the direction of travel kind of makes sense there as well but only if there was a similar unilateral distortion near bodies of mass.

And then I got to thinking about black holes and their extreme curvature - are they literally stopping universal expansion at their point of singularity such that deltaR=0 and therefore time stops? Or do they even stretch right back to R=0?

Ok, enough of my sillyness - can someone put this into perspective for me?

Answer 1

The universe isn't expanding through a fourth spatial dimension. It's common to visualise a curved manifold by embedding it onto a space of one higher dimension, but this is a purely mathematical operation to make it easier for our 3D brains to grasp. There is no sense in which the universe around us curves through another spatial dimension.

To try and make this clearer, imagine a universe with only one spatial dimension and model this by a piece of elastic with an ant walking along it. You could stretch and/or compress the elastic while the ant is walking and the ant would see some bits of spacetime expanding and shrinking. However everything is just in one dimension.

In mathematical terms, if you feel an urge to Google coming on, bending through an extra dimension is extrinsic curvature while in our elastic band model the deformation is intrinsic curvature. In GR, and as far as we know in real life, the curvature of the universe is intrinsic.

This distinction kind of invalidates the other questions you asked. However if you would like to repost your question in the light of my answer I'd be happy to attempt an answer.

Answer 2

It is best to only think of the expansion of the universe as happening over very large distance scales. Anything that is bound to something else, such as the way our finger is bound to our hand, or the way that Earth is bound to the Sun, is very decidedly not expanding, it is only when thing sare very far apart and the distance between them makes them essentially non-interacting that expansion starts to become a dominant effect.

Also, current observations say that the recollapsing spherical geometry is very unlikely to correspond with our physical universe.

Answer 3

It's a little hard to follow the specific question, but since you are beginning with introductory texts we will make the assumption that when you are referring to 4D, you are actually reading about 4 dimensional spacetime, where there are 3 dimensions of space and 1 dimension in time. I will refer you to this answer for a discussion as to the interpretation of dimension of spacetime. What is important is that when discussion relativity, we define a metric for that spacetime. In special relativity the metric is the Minkowski metric which is viewed as a representation of flat spacetime.

General relativity is the theory that deals with curvature of spacetime, where the metric of spacetime is not constant and can change depending on one's location in spacetime. Most experimental evidence however indicates that spacetime is flat globally, meaning it is flat everywhere, and the behavior of the universe is still best described the the $\Lambda CDM$ model which assumes the existence of flat spacetime, cold dark matter, and a cosmological constant. Based on this model, and experimental observation, the observable universe is 13.7 billion years old, and because relativity allows us to equate time to distance, we say that the "radius" of the observable universe is 13.7 billion light-years.

Based on observation, we have determined that the cosmological constant $\Lambda$ is near zero but slightly positive in value, this means that the universe is expanding and accelerating. This means that even though there may be enough mass to reach a critical density to stop the universe if there were no cosmological constant, because there is some amount of vacuum energy and space is increasing in size, the vacuum energy will eventually overpower the effects of gravity and the universe will expand and ultimately suffer a heat death.

As far as time stopping, relativity tells us that there is a relationship between objects traveling at different relative speeds, where an object that appears to move at relative speeds approaching light also appears to have its clock slow down, from the perspective of a stationary observer. This is called time dilation. In essence, as a person accelerates to the speed of light relative to the universe, their local clocks appear to be moving slower than those of an observer that has remained stationary. If a massive object where able to reach the speed of light, then its local clock would in principle completely stop.