### Section 1.4 - General Relativity

#### Newton's Law of Universal Gravitation

All matter is attracted to one another by a force, directly proportional to the product of the two masses and inversely proportional to the square of the distances between them
$F = G\frac{m_1 \cdot m_2}{r^2}$

#### General Relativity

Einstein recognized that special relativity was incomplete in and of itself because it only worked when objects moved at constant speeds. He wanted to generalize it to account for acceleration (what he called non-inertial frames of reference)

He realized that what we call gravity and what we call acceleration are the exact same thing. This was his famous though experiment:

From experience, we know that if we were in a free falling elevator, we'd experience weightlessness, so conversely, if the elevator were accelerating upward at a speed of $9.8 m/s$, you'd have no idea if you were in an accelerating elevator or just feeling the effects of Earth's gravity, 1G. Regardless, in both instances if you were to shoot a beam of light horizontally across the elevator, it would appear to bend from our perspective.

10 years after publishing his theory on special relativity, Einstein published his theory of general relativity in 1915, successfully reconciling special relativity with Newton's law of gravitation. General relativity is our best description of gravitation, defining gravity not as a force, but a distortion in the shape of spacetime.

The results of general relativity have many practical implications:

#### Distortion of Light

Light rays are bent by gravity. This is not something predicted by Newton's law of gravity because light has no mass, but this is resolved by Einstein's mass-energy equivalents

This can cause an effect known as gravitational lensing, where gravity from a supermassive object such as a galaxy or cluster of galaxies can amplify the light of something passing behind it. See video below from Wikipedia

#### Gravitational Time Dilation

An important premise of the movie Interstellar is that time runs slower where gravitational pull is slower. Here's why:
Once again, we know $speed = \frac{distance}{time}$. Imagine two beams of light, one traveling from A to B absent any gravitational pull (straight line), and one traveling from C to D through a strong gravitational field (curved spacetime)

The distance from C to D is longer due to the spacetime distortion, and since we know from the previous section that the speed of light is constant, time must slow down.

This has major implications for GPS and anything else that uses a super sensitive clock while orbiting the Earth. Gravity is slightly weaker in orbit than on the surface. Although the resulting time dilation is absolutely minuscule, it must be accounted for, otherwise clocks in orbit would be faster than clocks on the ground by 38 microseconds per day. GPS triangulation would become useless after 2 minutes, and after just one day, you'd be 10,000 kilometers off! (source)

Interestingly, both Einstein and Stephen Hawking noted that time must stock inside of a black hole, since light cannot escape. A strange implication of general relativity.

#### Gravitational Waves

If spacetime is like the fabric of the Universe, Einstein predicted that objects moving through spacetime should cause ripples in this fabric. If two massive enough objects collide, we should be able to observe these ripples. In 2015 (exactly 100 years after the publication of general relativity), the Laser Interferometer Gravitational-Wave Observatory (LIGO) annouced they had detected gravitational waves from a pair of merging black holes

LIGO works by shooting two perpendicular laser beams through 4km long tunnels and reflecting right back. If there were no disturbances, the light waves will match up precisely and cancel out. But if there was some ripple in spacetime, one tube will elongate and the other will shorten, causing a mismatch in wavelength that will be detected Great video demonstrating LIGO here

LIGO is so accurate it can measure a change in distance in its 4km tubes to 1/10,000th the width of a proton. That's equivalent to measuring the distance to the nearest star to the width of a human hair!