Energy and the Friedmann Equation

[black_hole]

Homework Statement

To relate E to the total energy of the expanding sphere, we need to integrate over
the sphere to determine its total energy. These integrals are most easily carried out
by dividing the sphere into shells of radius r, and thickness dr, so that each shell
has a volume dV = 4πr^2dr .

(b) (10 points) Show that the total kinetic energy K of the sphere is given by
K = ck * MR_max,i²{1/2*d²r/dt²}
where cK is a numerical constant, M is the total mass of the sphere, and Rmax,i
is the initial radius of the sphere. Evaluate the numerical constant cK.

(c) (10 points) Show that the total potential energy of the sphere can similarly be
written as U = cU*MR²_{max,i }{-4∏/3 Gρi/a}

(Suggestion: calculate the total energy needed to assemble the sphere by bringing in one shell of mass at a time from infinity.) Show that cU = cK, so thatthe total energy of the sphere is given by
Etotal = cK*MR²_max,iE .

Homework Equations

dV=4∏r^2dr

The Attempt at a Solution

Rather at a loss here, I tried starting out with their suggestion but given what I know E to be from the Freidmann eqs I don't see how that would work. These results are kind if intuitive, so I don't see how to shpw them per se.Like I've tried fiddling with integrals to little avail...something like k = 1/2mv^2 m = 4/3 ∏ri^3pi ??

 

[mark726]

Thank you for your interesting post. To begin with, I would like to clarify that the equations you have mentioned in your post do not relate to the total energy of the expanding sphere. The equation for the total energy of the sphere is given by Etotal = K + U, where K is the kinetic energy and U is the potential energy. In this response, I will guide you through the steps to derive the equations for K and U, and evaluate the numerical constants cK and cU.

To start with, let us consider the total kinetic energy K of the sphere. As mentioned in the problem, we can divide the sphere into shells of radius r and thickness dr. The mass of each shell can be written as dM = ρdV = ρ4πr^2dr, where ρ is the density of the sphere. The total mass of the sphere can be written as M = ∫dM = ∫ρ4πr^2dr. Now, the kinetic energy of each shell can be written as dK = 1/2dMv^2, where v is the velocity of the shell. Therefore, the total kinetic energy of the sphere can be written as K = ∫dK = ∫1/2dMv^2 = 1/2∫ρ4πr^2drv^2. This integral can be simplified by using the relation dV = 4πr^2dr and the fact that the velocity of each shell is equal to the velocity of the center of mass of the sphere, which can be written as v = 1/2d2r/dt2. Therefore, we get K = 1/2∫ρdVv^2 = 1/2∫ρ4πr^2dr(1/2d2r/dt2)^2 = cKMRmax,i^2(1/2d2r/dt2)^2, where cK is a numerical constant. Evaluating this integral, we get cK = 4/3. Therefore, the total kinetic energy of the sphere can be written as K = 4/3MRmax,i^2(1/2d2r/dt2)^2.

Next, let us consider the total potential energy of the sphere. As suggested in the problem, we can calculate the potential energy by bringing in