Relationship between mass and the radius of curvature of space and time

Question

What is the relationship between mass and the radius of curvature of space and time created due to the presence of the mass? please give the mathematical relation if there is any?

Answer 1

Spacetime is curved by more than mass, it is curved by any energy, and is curved by stress and pressure and momentum too. It can even be curved far away from any of those things, just in a different way. There is a natural way that spacetime can curve all on it's own, with the curvature in one place connected with the curvature of the places nearby and the curvature a little later influenced by the curvature earlier and a little farther away.

So first you should understand that kinds of curvature, that's what makes us go around the sun even though all the mass, energy, momentum, stress and pressure of the sun is way over there and we are over here. Then you can study how mass, energy, momentum, stress, and pressure allow spacetime to curve differently than that natural way.

Also keep in mind that time is part of the curvature too, in fact often it a big effect, sometimes the biggest, often near 50% of the cause of the effects you see. Even the deflection of starlight by the sun, the value that Einstein predicted was twice what a Newtonian based analysis would predict.

Finally as you've probably guessed, curvature is not just a simple radius like a curve in a plane, a whole wave of curvature can propagate through empty space. Or curvature can cooperate together to adjust itself around so as to not change in time, like the curvature far outside an ancient and isolated black hole that does not rotate.

Answer 2

What is the relationship between mass and the radius of curvature of space and time created due to the presence of the mass?

It is not that simple. The curvature of space-time does not only depend on a mass, like for example on the mass of the earth.

The curvature depends also on the distance from this mass. Near to the mass the curvature is larger (hence the radius of curvature is smaller). And far away from the mass the curvature is smaller (hence the radius of curvature is larger).

But in the Newtonian limit of general relativity (i.e. weak gravity and slow speeds) there is actually a very simple relation between the radius of curvature $R$ in 4-dimensional space-time and the gravitational acceleration $g$: $$R=\frac{c^2}{g}$$ You may find a derivation of this formula in my answer to another question.

Example:
At the surface of the earth we have $g=9.8\text{ m/s}^2$. This gives a radius of curvature $R=9.2\cdot 10^{15}\text{ m} \approx 1$ light-year.

Answer 3

The relationship between mass and the radius of curvature is encoded in the following equation:\begin{align}R_\mu=\frac{2G}{c^3}P_\mu\end{align} where $R_\mu$ is the component of the radius of curvature, $G$ is Newton's gravitational constant, $c$ the speed of light, and $P_\mu$ the component of momentum (derived by integrating Einstein's Field Equations over a 3-d hyper surface $dS^\nu$).

Momentum can be expressed such that:\begin{align}R_\mu=\frac{2Gm}{c^3}U_\mu\end{align} where $m$ is the mass and $U_\mu$ the relative velocity of the mass. If the mass is at rest then only the time-like component is non zero:\begin{align}R_0=\frac{2Gm}{c^2}\end{align} This may be recognized as the Schwarzschild radius. So the radius of curvature for a massive object at rest is equal to the Schwarzschild radius of a gravitational body of the same mass.

See "On the Fundamental Role of Massless Form of Matter in Physics: Quantum Gravity" by Alexander P. Klimets for derivations if needed. This paper lays this out beautifully in preparation of a quantum gravity framework and many other outstanding ideas.