# Spacetime line element to describe an expanding cube

1. Nov 24, 2012

### lailola

Hi, I have to write a spacetime line element for the shape of a cube of cosmological dimensions. This cube is expanding like this:

i)With time, the cube becomes elongated along the z-axis, and the square x-y shape doesn't change.

ii)The line element must be spatially homogeneus. (I dont know what this means).

I think there must appear the scale factor a(t) because of the expansion, but I don't know how to use the conditions of the expansion.

For a cilinder, I would use something like this: $dS^2=-dt^2+a^2(t)(R^2 d\theta^2+dz^2)$ where R is the radius of the cilinder.

Any help?

Thanks!

2. Nov 29, 2012

### clamtrox

Spatially homogeneous means that your universe is translation-invariant. In other words, the metric cannot depend on x,y or z.

If the cube gets elongated in the z-direction, then you need at least two scale factors: one for z and one for x and y.

3. Nov 29, 2012

### RUTA

$$ds^{2} = -c^{2}dt^{2} + dx^{2} + dy^{2} + a^{2}(t)dz^{2}$$
$$\dot a > 0$$

4. Nov 29, 2012

### lailola

Ruta, I don't get why that line element satisfies the first condition.

5. Nov 30, 2012

### clamtrox

This is certainly something you need to figure out before you can answer the question.

How would you know if something satisfies that condition? What does the condition mean, physically?

6. Nov 30, 2012

### lailola

It means that the area of the cube in the x-y plane is constant for every z. Doesn't it?

7. Nov 30, 2012

### RUTA

Do you understand comoving coordinates? Those in the z direction are being "stretched" while those of in x-y plane remain fixed.