Explaining Curved Space-Time to a Friend

[Berislav]

I see. I naively assumed that t' will become negative for some large value of tau.

 

[Chronos]

Mathematically, it can never be negative. Although it could be imaginary. Imaginary time is, however, hard to conceptualize.

 

[Berislav]

Mathematically, it can never be negative. Although it could be imaginary. Imaginary time is, however, hard to conceptualize.

I said that I assumed that t', (i.e,$\frac{dt}{d \tau}$ ) would eventually become negative even for a time-like geodesic. I never though that t itself would become negative, I thought that it would go to 0.

 

[pervect]

I got myself confused a while ago on a very similar issue. One thing I want to add in addition to the the specific solution I quoted for a specific expansion function a(t) is that we know in general that t' could not pass through zero for a timelike geodesic. This follows from the requirement of a timelike geodesic that

a^2(t) dx^2 - dt^2 < 0

after dividing both sides by d lambda^2 we get

a^2(t) (x')^2 - (t')^2 < 0

t' can never be zero and make the interval timelike, the best we could do is to make it lightlike (and that would be a boring, pointlike solution).

 

[Flatland]

Like what infinitetime said, how can the universe be finite and flat? That would mean the universe has an edge. What's beyond the edge? That would also mean the universe has a center which violates relativity because there is now an absolute reference frame. The only way the conditions exists where there is no absolute reference frame is either
1. the universe is infinite or 2. the universe is closed.