How Big Would a Black Hole Be if All the Stars in the Universe Combined?

[van gogh]

they say that if one black hole has two times more mass than the other black hole it's radius is two times bigger.
what if all the stars in the universe were to join to make one big big black hole how big would it be.
if a black hole with the mass of the sun is 3 km in radius and two times the mass of the sun is 6 km in radius then what would the black hole with all the stars in the universe be, what would it's radius be.
if there are 200 billion stars in a galaxy and there are 200 billion galaxy's in
the universe then there are 40,000,000,000,000,000,000,000 stars in the universe.
40,000,000,000,000,000,000,000 times 3 (because a black hole with a mass of a star the size of the sun is 3 km in radius) is 120,000,000,000,000,000,000,000.
so this black hole would be 120,000,000,000,000,000,000,000 km in radius
this is 20 billion light years in radius (because 1 light year is about 6,000,000,000,000 km)
now how is this possible if the universe is only 15 billion years old.
how can this black hole be bigger then the universe.?

 

[Garth]

To say that the "whole universe is a black hole" would be misleading, nevertheless you cannot get out of either so the general idea may not be too far off.

The BH is the spherically symmetric and static solution to Einstein's GR field equation, the cosmological solution is for the homogeneous density and non static situation. They are two different cases and must not be confused.

However the density of a Black Hole, dividing the mass by the volume, with r expressed in the 'standard' form of the metric, is given by, (with c = 1):

\rho_{BH} = \frac{3}{4\pi Gr^2}

and the critical 'closure' density of the universe is given by:

\rho_c = \frac{3H^2}{16\pi G}

if the universe's scale factor is

R(t) = R_0(\frac{t}{t_0})^n then H = nt^{-1} so

\rho_c = \frac{3n^2}{16\pi Gt^2}

so if the radius of the (observable) universe is taken to be r = t then

\rho_c = \frac{3n^2}{16\pi Gr^2}

and the last equation can be seen to be similar (of the same order of magnitude) to the first, with n = \frac23.

The numbers you worked out illustrate this.

Garth

 

[van gogh]

i was not trying to mislead anyone if i have i apologias and i did not say that the universe was a black hole i was just trying to imagine what it would be like if all the stars were to join to make one big big black hole.
i appreciate you clarifying my question.
i am not a physicist or any thing like that i just like to read about universe and evrey thing in it. now if a had understood my question i would not have post it here, so again if i mislead anyone i am sorry

 

[Garth]

i was not trying to mislead anyone if i have i apologias and i did not say that the universe was a black hole i was just trying to imagine what it would be like if all the stars were to join to make one big big black hole.
i appreciate you clarifying my question.
i am not a physicist or any thing like that i just like to read about universe and evrey thing in it. now if a had understood my question i would not have post it here, so again if i mislead anyone i am sorry

Not at all van gogh! It was a good question, keep them coming.

And welcome to these Forums!

(But it might have been more appropriate to have posted it in the Special & General Relativity Forum)

Garth

 

[van gogh]

well thanks and i do have one question is space something like rubber that stretches around a massive object am just trying to have an image of it.

 

[Garth]

Yes van gogh, a standard analogy of the gravitational field around the Sun, for example, is to consider a flat rubber sheet. Small balls would roll across it along straight lines, but if now a bowling ball is placed in the middle so that it makes a dip, then the small balls would be deflected towards the centre - the Sun - or they can even be made to 'orbit' around the circumference of the dip.

As an analogy this model can be very helpful but realize that it is only an analogy and breaks down in that it uses gravity twice, to keep the balls on the sheet whilst trying thereby to explain gravity! The real situation is to visualise the curvature of the rubber sheet intrinsically. The paths of the small balls are now straight lines across the curved surface of the sheet, their paths are bent, or even orbit in a closed ellipse, because the surface on which they are 'drawn' is curved.

This sheet represents curved space, in GR gravitation is properly described by the curvature of space-time. The time component is not 'curved' instead it is revealed as time dilation.

I hope this helps.

Garth

 

[van gogh]

thank you, yes that helped