One of the most important realizations Einstein made while developing special relativity is that there is no such thing as a universal time or distance. Instead, special relativity introduced the notion of an invariant spacetime interval. When astronauts travel at different speeds and experience differences in the duration of time intervals, they are effectively trading distance for time or vice versa. They are experiencing a distortion or warping of space and time together. This is required in order for all observers to agree on the speed of light. Einstein realized that he could explain the effects of gravity by combining the equivalence principle with the concept of an invariant spacetime interval used in special relativity. When Einstein developed the general theory of relativity, he came to the realization that gravity is the warping of spacetime. So, stars like our Sun, which have strong gravitational field due to their large mass, actually bend and stretch the fabric of the universe itself. The warping of spacetime causes planets and light to travel on curved paths near massive objects. An early test of general relativity depended on the bent spacetime around the Sun. In 1919, the astronomer, Sir Arthur Eddington, led an expedition to an island of the West Coast of Africa in order to measure how much the gravity of the Sun warped spacetime. They did it by observing a total eclipse of the Sun. During the eclipse, a star could be seen next to the eclipse Sun. Those of you who have observed a total solar eclipse, like I have, know that the Sun looks eerily like a black hole in the sky surrounded by white hair. The locations of all the stars in our neighborhood of the galaxy were mapped more than 100 years ago. So it was known that this star should really be located behind the Sun when viewed from the Earth on the day of the eclipse. So, how did Eddington and his team see the star? The light from the star traveled on a curved path around the Sun to the astronomers' telescopes. This effect was predicted by Einstein's theory of general relativity. The measurement they made confirmed and has since been reconfirmed that the theory of general relativity is accurate and that spacetime itself is bent by matter. This changes our notion of what a straight line is, of course, because if spacetime itself is bent, how can we possibly know that we're going in a straight line? Instead of calling them straight lines, in general relativity, we call them geodesics, which represent the straightest possible path of an object in a curved spacetime. Even though geodesics represent straight lines in curved spacetimes, they wouldn't be considered straight by our standards. Just as the Sun's mass bends the spacetime around it, any light crossing bent spacetime will appear to have its path bent. Since we are talking about curving spacetime, let's consider the surface of this chalk ball as a section of a curved two-dimensional space. This works equally well if you imagine the chalk ball to be the Earth. If I asked you to draw a straight line between two points on opposite sides of the ball, the same thing as asking for the flight path between two cities on Earth, you might be tempted to draw along an equatorial line to join them together. Even though I asked you to draw a straight line, already it's curved. Instead, the smallest distance between two points on a curve surface is considered straight if it's also the shortest line joining the two points together. The smallest distance between two points on a curved surface is called a geodesic. If you look at the flight path of an airplane from Toronto to London, the airplane crosses the ocean near Greenland. The shortest route joining the two cities is a curved path. The same is true for any object traveling through curved spacetime. Now, where do you think the most convoluted curvatures of spacetime in the universe exist? That's right, black holes. Not only do black holes warp spacetime, they warp it to the point that even light will travel on highly curved paths. Photons, by definition, travel on geodesic paths in spacetime. Close to the black hole, the curvature becomes so high that light is bent into paths that all terminate at the black hole singularity. General relativity interprets gravity as the warping of spacetime. When we view a picture of the gravitational field around a massive object, it's usually represented as a depression in space. However, we need to understand that gravity also warps the passage of time. It's strangely difficult for the human mind to grasp the concept of warping spacetime. We understand what it means to bend or warp a material like plastic, but what does it mean when the actual space and time that we live in are bent and twisted? In a sense, warped spacetime means that the paths we choose to cross space and time will be shorter or longer in distance between two points and in the duration it takes to travel between them depending on what the gravitational fields are along the path. Let's focus specifically on how gravitational fields warp the time component in an effect called gravitational time dilation. Let's start with an example by considering two astronauts exploring an unstudied planet around a distant star, perhaps planet e in the nearby Trappist-1 System, which we'll shorten to trappy. One astronaut needs to stay with the ship in order to orbit around the parent star while the astronaut descends to trappy surface. Since we are talking about time, both astronauts will need to carry clocks, which they synchronize before they separate. Far from the surface of the planet, both clocks tick in perfect synchronicity. One astronaut now descends to the surface of trappy. On the surface, he is deep in the planet's gravitational well and therefore, experiences a greater gravitational force. The spacetime in the vicinity of the planet will also be warped. The effect that the warping has on the astronauts' clocks causes it to tick more slowly than the one in orbit. For every tick of the clock on the surface, the orbital clock ticks more rapidly. On the surface of trappy, the astronaut doesn't experience the change in the passage of time because all biological processes are likewise slowed down by the warping of gravity. Just like the ticks of the clock, a distant observer would see the heartbeat of an astronaut on the surface to beat more slowly. Once the surface mission is complete, the two astronauts rejoin one another in orbit around trappy. The astronaut who stayed in orbit will be dismayed. She experienced a longer time than the astronaut who was on the surface. Depending on the duration of the stay and the strength of the gravitational field, the astronaut who went down to the planet's surface will experience fewer ticks of the clock and therefore, be several seconds younger than the one who stayed in orbit. To calculate how time has worked in a strong gravitational field, the following equation is employed. Delta t planet, the elapsed time on the surface of the planet, is equal to Delta t orbit, the elapsed time on the orbiting spaceship, times the square root of one minus two times G times mass divided by radius times c squared. In this formula, the mass and radius refer to the mass and radius of the planet. But if instead of a planet, you were a distance R from a star or a black hole with mass M, you could use the same formula. The important thing in this formula is that the quantity inside the square root sign is smaller than one. So the amount of time that passes when you're in a gravitational well is smaller than if you're out in space far from the gravitating object. Note that this formula doesn't make sense if the ratio of the mass to radius gets too large. This formula only makes sense if R is larger than two times G times mass divided by c squared. You might think that your everyday life is not much affected by time dilation due to special or general relativity. However, you may be surprised to learn that almost everyone carries a piece of technology that would be useless without both theories, GPS. The Global Positioning System that you use every time you navigate with a map on your smartphone depends on Einstein's theory of relativity to function correctly. Handheld GPS works because the device inside your smartphone is capable of measuring and comparing the signals from multiple satellites in orbit around the Earth. These satellites are placed in well-known orbits and carry very precise clocks. By broadcasting a timing signal that can be picked up on a GPS receiver, the difference in timing signals from different satellites can be used to triangulate your position. Since GPS satellite travel at about 14,000 kilometers per hour, they experience a very slight time dilation due to special relativity. Each day, a satellite's clock would appear to slow down by about seven microseconds. That doesn't sound like much, but if you neglected this drift, your GPS would accumulate an error of about two kilometers every day. General relativity predicts that the clocks aboard a GPS satellite traveling at an altitude of 20,000 kilometers would appear to tick faster than clocks on Earth. Every day, a satellite clock would appear to speed up by 45 microseconds compared to clocks on Earth's surface. If this error wasn't corrected, the GPS would accumulate an error of over 13 kilometers a day. Since special relativity works to slow down the apparent rates of the clock on a GPS satellite, and general relativity speeds up their apparent rates, the combined effects add up to a 38 microseconds per day error. Without relativity, our GPS devices would drift by over 10 kilometers every day, roughly the same as 12 centimeters a second. Luckily, we know about the effects of relativity. So we can correct for this drift. GPS devices are some of the most robust tests we have for Einstein's theories of relativity. In the movie "Interstellar", the main character Cooper is sent to retrieve a fellow explorer Mann from the surface of Miller's planet orbiting the nearby black hole Gargantua. On the surface of Miller's planet, an hour of time is equivalent to seven years on Earth. This is an example of the correct use of an effect called gravitational time dilation. Since gravitational time dilation slows down the passage of time in intensely strong gravitational fields, it equally effects physical processes that are time-dependent. That means that someone observing an astronaut in orbit around the black hole would see their clocks ticking slow and their hearts beating slower, and everything about them slow down. So, what happens to a beam of light when it's generated deep in the gravity well near a black hole? The beam of light experiences gravitational redshift. Recall the Doppler effect that we discussed earlier. When a moving object like a rocket ship is emitting light, the light can be blueshifted or redshifted depending on the ship's motion towards or away from the observer. If a ship were to accelerate away from you, you would see the light from its engines becoming redder and redder as it accelerated to ever increasing speeds. Light emitted from deep within gravitational well has to work against gravity in order to leave a planet, or a star, or the region near a black hole. When light travels away from a planet, the photon has to convert kinetic energy into gravitational potential energy. If we remember that red photons have less energy than blue photons, we can predict that the photons emitted from the surface of a star will appear redder to an observer far from the star. This effect is called gravitational redshift. The gravitational redshift effect is very small, but it has been measured in the light emitted by a white dwarf star and it agrees with the predictions in general relativity.