Spacetime line element to describe an expanding cube

In summary, the cube of cosmological dimensions is expanding with time, elongated along the z-axis while the x-y shape remains unchanged. The line element must be spatially homogeneous, meaning it cannot depend on x, y, or z. A possible solution for the line element is ds^{2} = -c^{2}dt^{2} + dx^{2} + dy^{2} + a^{2}(t)dz^{2} where a(t) is the scale factor for the expansion. However, further understanding of comoving coordinates is necessary to fully understand and satisfy the first condition.
  • #1
lailola
46
0
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: [itex]dS^2=-dt^2+a^2(t)(R^2 d\theta^2+dz^2)[/itex] where R is the radius of the cilinder.

Any help?

Thanks!
 
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  • #2
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
lailola said:
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: [itex]dS^2=-dt^2+a^2(t)(R^2 d\theta^2+dz^2)[/itex] where R is the radius of the cilinder.

Any help?

Thanks!

[tex]ds^{2} = -c^{2}dt^{2} + dx^{2} + dy^{2} + a^{2}(t)dz^{2}[/tex]
[tex]\dot a > 0[/tex]
 
  • #4
RUTA said:
[tex]ds^{2} = -c^{2}dt^{2} + dx^{2} + dy^{2} + a^{2}(t)dz^{2}[/tex]
[tex]\dot a > 0[/tex]

Thanks for your answers.

Ruta, I don't get why that line element satisfies the first condition.
 
  • #5
lailola said:
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?
 
  • #6
clamtrox said:
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?
 
  • #7
lailola said:
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.
 

1. What is a spacetime line element?

A spacetime line element is a mathematical expression used in Einstein's theory of general relativity to describe the geometry of spacetime. It is a way of representing the relationship between space and time in a four-dimensional space.

2. How is a spacetime line element used to describe an expanding cube?

In the context of an expanding cube, the spacetime line element is used to describe the relationship between the dimensions of the cube as it expands and the passage of time. It takes into account the changing size and shape of the cube as it expands, and how this relates to the concept of time.

3. What is the significance of using a spacetime line element to describe an expanding cube?

Using a spacetime line element allows us to understand the expansion of a cube in the context of Einstein's theory of general relativity, which provides a more comprehensive understanding of the relationship between space and time in the universe. It also allows for accurate predictions and calculations related to the expansion of the cube.

4. How is a spacetime line element different from a traditional geometric formula?

A traditional geometric formula only describes the relationship between the dimensions of an object in a three-dimensional space. A spacetime line element, on the other hand, takes into account the effects of time and the changing size and shape of the object in a four-dimensional space.

5. Can a spacetime line element be used to describe other expanding objects in the universe?

Yes, a spacetime line element can be used to describe the expansion of any object in the universe, as long as it takes into account the four dimensions of space and time. It is a fundamental concept in Einstein's theory of general relativity and is applicable to various phenomena in the universe.

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