How is time incorporated into the robertson walker metric?

[NihalRi]

I watched a lecture that derived the robertson walker metric by creating a metric to describe a four dimensional sphere in three dimensions. Then from minkowski's equation-

https://lh3.googleusercontent.com/proxy/dWewIZYdx7aQRy2DtIjNF60tzgEdwxmrbyaqHl1QsAGyrENiErbEKLdBESnKijWbHk0UAnK4tRxkvi2Uiz-ZGUV7N3560lx8wdKofZY-YALhDORrks_t80D5gX38Y8Vtcf2lDg=w252-h22-nc
the time variable was added.

And this equation came by using pythagorus' theorem on the minkowskian plane. So my question is, why is there a minus sign after the time variable? I would have thought is to be a plus.

The final equation became-

 

[_PJ_]

Time is in the negative direction in Minkowski as with GR.

You could always instead give three spatial dimensions as negative, with positive signs, but whichever way, the 'metric signature' will always have time in the opposite sign to the spatial dimensions.

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It used to be a lot more difficult to deal with when considered as time axis having an imaginary component:
https://en.wikipedia.org/wiki/Wick_rotation

 

[NihalRi]

Interesting, Does negetive time imply anything?

 

[Jorrie]

Time is in the negative direction in Minkowski as with GR.

You could always instead give three spatial dimensions as negative, with positive signs, but whichever way, the 'metric signature' will always have time in the opposite sign to the spatial dimensions.

Interesting, Does negetive time imply anything?

It is a property of the spacetime interval, which is based upon the difference between the squared time interval and the squared space interval between events. Negative time per se is not involved. Which one is subtracted depends on the type of interval: timelike intervals have the space interval after the negative sign and vice versa for spacelike intervals.

 

[NihalRi]

I might be digressing but I had thought spacelike and timelike referred to situations where there isn't a change in time and when the isn't a change in space respectively. Could it be that there is a boundary between the two(lightlike?) and our universe is leaning towards timelike. And does this have anything to do with how our universe is open? With an open universe I'd assume the space increases faster than time and that the robertson walker would have the time interval after the minus sign which clearly isn't the case so I'm confused.

 

[NihalRi]

Oh and I also read something about the value of k determining if our universe is open or closed how does that fall into all this.

 

[Jorrie]

I might be digressing but I had thought spacelike and timelike referred to situations where there isn't a change in time and when the isn't a change in space respectively. Could it be that there is a boundary between the two(lightlike?) and our universe is leaning towards timelike.

No, timelike spacetime intervals mean that the time interval between two events is larger than the space interval, i.e any interval where ##(c\Delta T)^2 > (\Delta X)^2 ## for flat spacetime. It also means that a material body can move between the two events at v<c. The opposite is true for spacelike intervals and lightlike intervals occur, as you suggested, when the two are equal and only light can move between the two events. For curved spacetime like our universe, the situation is a little more complex, but the same principle applies. But the universe is not timelike or spacelike: the terms apply only to events in spacetime.

Oh and I also read something about the value of k determining if our universe is open or closed how does that fall into all this.

Yes, 'k' in the RW metric is the curvature parameter. As you can see, it only affects the a(t) part of the RW metric, but the expansion factor a(t) is obviously a function of time, hence it represents spacetime curvature. When k=0, you have the Minkowski metric, just expressed in polar coordinates, not Cartesian as in you first equation.