1.2 Einstein’s Gravity

summary

For many years Newton's theory of gravity was canonical.

- Newton’s theory was revolutionary because it mathematically described the force of gravity between masses.
- His theory explained every motion people knew about at the time, bridging Galileo’s observations of falling bodies at the terrestrial scale with Kepler’s laws of planetary motion and unifying them in the law of universal gravitation.
- The law of universal gravitation itself is actually very simple. It states that the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them:$$ F = \frac{Gm_{1}m_{2}}{r^2}$$

Newton’s theory clashed with Einstein’s special relativity.

- One of Einstein’s motivations for proposing general relativity was his work on special relativity. He had postulated that there is a “universal speed limit,” the speed of light, which is the maximum speed that any object could have.
- In Newton’s theory the force of gravity is transmitted instantaneously. If the Sun moves, the Earth experiences the change immediately.
- This clearly violated Einstein’s speed limit and sent him searching for a better understanding of gravity.

Einstein proposed general relativity and the concept of spacetime.

- General relativity does not provide a formula for the force between two masses. In fact it says that there is no such force at all!
- Instead, masses exist in and interact with a dynamic spacetime. Distortions in spacetime create the effects perceived as gravitational force.
- When masses move they change the fabric of spacetime around themselves, causing other masses to move in turn. A mass in orbit is not being held by a force, but is actually following the shortest path (called a
**geodesic**) through a warped spacetime. - Changes in spacetime propagate at a finite speed, the speed of light, avoiding any violations of the universal speed limit.
- Einstein put the description of the geometry of spacetime in the Einstein tensor, a four by four matrix that combines the three dimensions of space with time (hence the name). That geometry ($G_{\mu v}$) depends on the mass and energy ($T_{\mu v}$) in a given region of spacetime according to the Einstein field equation: $$G_{\mu v} = \frac{8 \pi G}{c^4} T_{\mu v}$$

After creating his theory, Einstein looked for testable predictions.

- General relativity predicts that the universe expands. Einstein did not like this notion and added a constant to his equations to make the universe static, though it was eventually observed that the universe is actually expanding.
- Existence of black holes is another prediction. The first exact solution to Einstein’s equations was not done by Einstein but by Karl Schwarzschild, and it suggested the possibility of a singularity or black hole. Einstein did not like this either, though we now believe that black holes exist.
- The one prediction that Einstein really liked, and the one that brought widespread acceptance for general relativity, was that light bends when it passes near large masses like the Sun. This effect, called gravitational lensing, was observed during a solar eclipse in 1919 and made headlines worldwide.
- Another interesting prediction is that time passes at different rates at different heights. This is important not only as a testable prediction of general relativity but also because it influences technology like GPS. Without correcting for the asynchrony of clocks at different altitudes GPS would not work properly.
- A prediction that was very different from Newton’s theory of gravity is the existence of gravitational waves. While Newton says that two stars will orbit each other forever, Einstein predicted that the two stars will produce spacetime distortions that actually travel away from the system, carrying energy. This lost energy will cause the stars to get closer together and eventually collide, a very different fate than Newton predicted.