Einstein's Theory of Gravity

Newton's Law of Universal Gravitation treats gravity as an instantaneous action-at-a-distance force. Einstein's General Relativity replaces this picture with a geometric description of 4D spacetime.

The Principle of Equivalence

There is no physical experiment that can distinguish between a uniform gravitational field $g$ and a frame of reference accelerating uniformly at $a = g$ in deep space.

As a consequence, an object in free fall does not experience a force; it is weightless and following a natural trajectory through curved spacetime.

Gravity as Spacetime Curvature

Mass and energy do not exert an instantaneous pulling force across space; instead, they distort the geometry of spacetime around them:

Geodesics: Objects (and light) moving through space follow the straightest possible paths through curved spacetime.
Predictions: General Relativity successfully explains phenomena where Newtonian gravity fails, such as the precession of Mercury's orbit and the gravitational bending of light.

Black Holes and the Schwarzschild Radius

When a mass $M$ is compressed inside a small enough region, the spacetime curvature becomes so intense that the escape velocity equals the speed of light $c$ .

The radius defining this point-of-no-return (the event horizon) is the Schwarzschild Radius $R_S$ :

R_S = \frac{2GM}{c^2}

(1)

Example 1

To what radius would Earth ( $M_E = 5.97 \cdot 10^{24} \text{ kg}$ ) need to be compressed to become a black hole?

Hint Click to unblur

Use Equation 1 with the speed of light $c = 3.00 \cdot 10^8 \text{ m/s}$ .

Solution Click to unblur

Using Equation 1:

R_S = \frac{2GM_E}{c^2}

Substitute the known values:

R_S = \frac{2 \left(6.674 \cdot 10^{-11} \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2}\right) \left(5.97 \cdot 10^{24} \text{ kg}\right)}{\left(3.00 \cdot 10^8 \text{ m/s}\right)^2}

R_S \approx 8.85 \cdot 10^{-3} \text{ m} \quad (8.85 \text{ mm})

If Earth were compressed to roughly the size of a marble, it would become a black hole.