Dear Friedman

We are much smarter than hundred years ago.

The GR, the Schwarzschild metric, gravitational field, spherically symmetric object is invariant under everything including time and time reversal.

It is time to think out the box, and put all these mechanical theory that was pronounced over 100yrs ago for one galaxy universe behind.

Static metric describing gravitational field of spherically symmetric symmetric object can be invariant under time reversal only theoretically. Spaces with direct and reverse flow of time belong to in principle different objects. It is true for any metric describing physical object including creating spherically symmetric gravitational field. We perceive the observed time as flowing from the past to the future. Until we live in material world, we perceive only direct flow of time. Spaces with reverse flow of time are exotic for us. One say, person at the moment f death perceive the time as flowing from the present (death) to the initial moment of the life. It is interesting to note that persons remembering past reincarnations perceive the time s flowing to the future.

Now return to our physic, which is based until on space-time. Components of every static metric including spherically symmetric do not depend from coordinate time t = x^0/c , but for every observer of space-time the observable time exists. Static space does not rotate (all components g_0i = 0, i = 1,2,3). (For stationary metrics some or all g_0i are not zeroes).

That to understand what is it reverse or stopped time it is necessary to use notion "observed time tau". For static metrics interval of observed time is d/tau = (g_00)^1/2 dt, where t is coordinate (ideal) time. Example: for well known Schwarzschield metric d'tau = (1 - r_g/r)dt, r_g = 2GM/c^2. The time flows frm the past to the future by r>r_g, stopped by r = r_ and possess reverse direction by r

Stefan,

The physical problem implies only that the metric describing the field is independent on time, but it does not exclude to have a non-zero term f(r)dtdr, In such case, unlike in Schwarzschild universe, light propagating toward the massive object will not slow down.

Javad,

As you suggested, I will publish soon a different solution for such field which keeps all spacetime symmetries of the problem, propagate with the speed of light and satisfy the Newtonian limit. These 3 properties define uniquely such metric without the need for Einstein's field equation. The dynamics based on this metric passes all classical tests of GR, but the metric is not invariant under the time reversal. It leads to new predictions, not tested yet.

Larissa,

The answer to Javad can partially answer also your understandings.

Dear Stefan,

I am aware of this coordinate transformation that you mentioned. However, what is the physical interpretation of the new time and radial coordinates?

I have a new paper Article Relativistic Gravitation Based on Symmetry