Spacetime line element to describe an expanding cube

[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 don't 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!

 

[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.

 

[RUTA]

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 don't 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!

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

 

[lailola]

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

Thanks for your answers.

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

 

[clamtrox]

Thanks for your answers.

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

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?

 

[lailola]

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?

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

 

[RUTA]

Thanks for your answers.

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

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