If a 4th dimension existed would universe expand anisotropically?

[jk22]

Suppose the universe were described by internal geometry by a ball, i.e. the metric where :

$$diag(1,r^2,r^2 sin(\theta)^2)$$

Now if we go to exterior geometry and suppose there existed a 4th timelike dimension the manifold were for example modelized by :

$$\left(\begin{array}{c} x=r\sin(\theta)\cos(\phi)\\y=r\sin(\theta)sin(\phi)\\z=r\cos(\theta)ch(\alpha)\\w=r\cos(\theta)sh(\alpha)\end{array}\right)$$

From the 4th dimension we could extract the time by assuming isotropy : $ct=r sh(\alpha)\Rightarrow z=r\sqrt{\frac{c^2t^2}{r^2}+1}\cos{\theta}$

However this becomes mathematically incorrect since now $\alpha$ is a function of r and has to be considered so to compute the metric out of the vector field, but could this give an idea on how behaves the universe at large scale, namely an anisotropic accelerated expansion ?

 

[anuttarasammyak]

I am not sure you mean (3+1) space with space z and time w is abnormal or (4+1) space with 4 x,y,z,w are all space components of
$$dl^2=dx^2+dy^2+dz^2-dw^2$$?
Show us metric of your world for confirmation.

 

[PeterDonis]

suppose there existed a 4th timelike dimension

We cannot discuss personal speculations here.

Thread closed.