Because this ball, independently of its radius,

creates the same Coulomb field outside of its body.

We have nonzero electric field but zero magnetic field.

As a result, there is no radiation.

Despite the fact that during this shrinking,

there is an accelerative motion of electric charge.

Despite the fact that we have an accelerative motion of

the electric charge.

There is no radiation in such an ideal situation.

The reason for that is the following.

That to have electromagnetic radiation,

one has to have at least dipole moment changing in time.

While for the ideal spherical symmetry, not only dipole, but all the momenta.

All the multiple momenta with respect to the center of the ball as zero.

With respect to the center of the ball as zero.

So this is a situation with electromagnetic radiation.

With the gravitational radiation, the situation is even more refined in

the sense, well first of all, there is no radiation also.

Even can be guessed within Newtonian approximation.

Because independently of the radius of such a ball.

Outside of it, it would create a Newtonian limit, just Newtonian potential.

Independently of the same Newtonian potential corresponding to the same mass.

As if it is situated in the very center of the ball.

So this is the essence,

again, of some kind of revelation of the Birkhoff's theorem.

That independently of the size of this guy,

if the spherical symmetry is never violated.

We have outside of the ball the same Schwarzschild metric with fixed mass.

Intuitively should be understood that if there is no energy fluxing out.

The total energy of this ball remains constant.

And the total energy of this ball is just its mass.

Okay, that's the situation we have.

But at the same time, the surface, the radius

of this ball is the function of time.

The radius of the surface of this ball is a function of time.

So we assume that the world-surface of the surface

of the ball is as follows.

It's the following vector.

So it's [T(tau), sigma(tau)].

And we should stress that throughout this lecture.

We assume that we denote by the same sigma.

By the same sigma, we denote two different things.

Both the world-surface of the surface and the surface also.

So world-surface And

surface of the bowl at the same moment.

I hope it will not cause confusion.

So, in spherical code and this guy occupies all the angles theta and

phi, which defined this metric, and

R as a function of proper time at the initial moment of time t0 is R0.