Interpreting Schwarzschild coordinates

[Mentz114]

Given that the Newtonian 'radius of escape' for light is given by the relation
$$\frac{1}{2}v^2_{esc}=\frac{GM}{R}<\frac{1}{2}c^2$$
from which we get the same value as the Schwarzschild critical radius,
$$R_{crit}=\frac{2GM}{c^2}$$
why can't we identify R with the Schwarzschild coordinate r ?

The deriviation of the Schwarzschild metric starts with the line element in R, T, \theta, \phi and after symmetry constraints there is still a dTdR term. This removed by a transformation between Schwarzschild r,t and T which mixes space and time like a boost. This says that r is not R. This whole procedure looks suspect (to me) because there is no Newtonian line element that has time in it.

We know now that the dRdT coefficient will be zero for a non-rotating source, so we could start from a spherically symmetric metric with only diagonal terms, solve the field equations and arrive at the Schwarzschild coords without the space-time mixing transformation that got rid of the off-diagonal element. In fact without any transformations at all.

 

[Philosophaie]

$$R_{crit}$$ is fixed for each object (ignoring all the weak gravitational effects from other objects). The Schwarzschild Radius for Earth is about 9mm. The dTdR term means that there is not a pure gravitational field. There must be a strong magnetic or electric field associated with the object in question. If the object in question is the sun the magnetic field filps on a regular basis and there is a question of the solar wind phenomena carrying added magnetic field through the solar system.

 

[Entropia]

Interpreting Schwarzschild coordinates can be a complex task, as it involves understanding the underlying principles of general relativity and how it differs from Newtonian physics. The Newtonian 'radius of escape' for light, as given by the relation \frac{1}{2}v^2_{esc}=\frac{GM}{R}<\frac{1}{2}c^2, is a concept that is only applicable in the realm of Newtonian physics. In general relativity, the concept of escape velocity is not as straightforward, as the curvature of spacetime plays a significant role in the motion of objects.

The Schwarzschild critical radius, R_{crit}=\frac{2GM}{c^2}, is a fundamental value in general relativity, as it represents the point at which the curvature of spacetime becomes infinite. This value is derived from the Schwarzschild metric, which is a solution to the Einstein field equations that describe the curvature of spacetime in the presence of a non-rotating, spherically symmetric mass.

It is important to note that the Schwarzschild metric is derived from the assumption that the spacetime is spherically symmetric, and therefore, the resulting coordinates (r, t) are also spherically symmetric. This means that the coordinate r is not equivalent to the Newtonian radius R, as they are derived from different assumptions and principles.

Furthermore, the transformation between the Schwarzschild coordinates and the Newtonian coordinates is not a simple boost, but rather a complex transformation that takes into account the curvature of spacetime. This is necessary in order to accurately describe the motion of objects in the presence of a massive object.

In general relativity, time is not treated as a separate entity from space, but rather as a component of spacetime. This means that the line element in general relativity will always involve both space and time, unlike in Newtonian physics where time is treated as a separate dimension.

In conclusion, the inability to identify R with the Schwarzschild coordinate r is due to the fundamental differences between Newtonian physics and general relativity. The Schwarzschild coordinates are derived from the principles of general relativity, which take into account the curvature of spacetime, while the Newtonian radius of escape is a concept that is only applicable in the realm of Newtonian physics.