Mathamatical Description of the Cosmological Principle

[Haths]

I think this belongs here, while not relativistic, is is about objects moving relative to one another.

Summery:
This stems from the description of 'Dark Energy' in a mathamatical context to show that you can place a 'center of the universe' at any point in space, and the accelaration vector between two bodies is inderpendant of the position vectors to this 'arbitary center of the universe'.

Body:
The presumption is that there exsits a 'center of the universe', and that from it we can draw a diagram


A
...\
...\___________________
...B......C

Where A is our 'center of the universe', B is one body, C is the other body. There is a radial distances r_B and r_C respectfully, and the magnitude of radial accelaration from this 'center of the universe' is proportional to these distances.

From B, an oberserver notes that C is accelerating away from him at some accelaration a'_C and reversely, from C an observer notes that B is accelerating away from him at some accelaration a'_B

How do I show that no matter where I place A, the magnitude of accelaration is not dependant on the 'center of the universe'?

I thought you could describe a'_C and a'_B as;

a'_C = r_CK*cos( 1/2 n ) - r_BK
a'_B = r_BK*cos( 1/2 n ) - r_CK

Where K is the constant of proportionality and n is the angle between the bodies. Using the geometry between them. I guess that the cosine formula is now somehow incorperated into this;

a²=b²+c²-2bc*CosA

Because there are simmilarities in both equations, but I'm not sure where to go from here in the proof...

Haths

 

[Mentz114]

You can model the principles of isotropy and homogeneity by setting up a line element that defines the distance measure in the cosmos by all observers. By ansatz we say that every observer will be able to construct a spherically symmetric coordinate system around themselves. Spherically symmetric implies no preferred direction (isotropy) and all observers can do this ( homogeneity).

The line element is one of the Robertson-Walker-Lemaitre varieties and the simplest form looks like this

$$ds^2 = c^2dt^2-a(t)^2d\sigma^2$$

see Wiki on 'Robertson-Walker metric'.

 

[Entropia]

ara,

The Cosmological Principle is a fundamental concept in cosmology that states that the universe is homogeneous and isotropic on a large scale. This means that on average, the distribution of matter and energy is the same in all directions and at all points in the universe. This principle is often used in the study of dark energy, as you mentioned, to show that the acceleration of objects is independent of an arbitrary center of the universe.

To mathematically describe this principle, we can use the concept of radial acceleration, as you have mentioned. Let us consider two bodies, B and C, with radial distances rB and rC from an arbitrary center A. The magnitude of radial acceleration from A can be represented as a function of these distances, denoted as aB and aC, respectively.

Now, let us consider an observer at B who measures the acceleration of C as a'C. Similarly, an observer at C measures the acceleration of B as a'B. As you have correctly pointed out, we can represent these accelerations as:

a'C = rCK*cos( 1/2 n ) - rBK
a'B = rBK*cos( 1/2 n ) - rCK

where K is the constant of proportionality and n is the angle between the bodies. However, we can also use the cosine formula to rewrite these equations as:

a'C = √(rCK^2 + rBK^2 - 2rCKrBK*cos(n))
a'B = √(rCK^2 + rBK^2 - 2rCKrBK*cos(n))

Notice that both equations have the same form, with the only difference being the order of the terms. This means that no matter where we place the arbitrary center A, the magnitude of acceleration will remain the same, as it is solely dependent on the radial distances and the angle between the bodies.

In other words, the Cosmological Principle holds true as the acceleration of objects is independent of the position of the arbitrary center of the universe. This mathematical representation helps us understand that the universe is uniform on a large scale, and there is no preferred reference point or center.

I hope this helps in understanding the mathematical description of the Cosmological Principle. Keep exploring and questioning!