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Space and Time Part I: Gravity and Curved SpaceTime by Mark Lawrence 
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Imagine you're suddenly alone in an empty universe. There's just you, no light, no dust, nothing but you and black emptiness. This is what we might imagine for just before God said "Let there be light."
Now let's ask a few questions about this. Can you measure the distance between two points? Interestingly, the answer is no. We have a strong intuition that space and time are like a stage, and all the atoms and light rays and other thingies are actors upon this stage. When I said an empty universe, I imagine you thought of a dark empty room. But this intuition is wrong. If there is no matter, there is no space. We'll consider this in a few different ways and try to understand it.
Back to our question: can you measure the distance between two points? How would you do this? Normally, you would hold up a yard stick and see how far apart the two points are. This works for a rather odd reason: all atoms are about the same size. When you line up 10 billion atoms, you have a line of atoms that's just about exactly one meter long, or one yard long if you prefer yards. To measure distance, we count off atoms  ten million atoms is a millimeter, about a 25th of an inch. But in our empty universe, there are no atoms, so there's no yard stick.
Another popular way to measure distance is to use light. Radio waves have a wavelength of about a meter, so we can use a radio wave to measure distance  we just count off wavelengths and that equals meters. If we want to be more precise, we can use visible light. Visible light has a wavelength of about a millionth of a meter, so a million wavelengths equals one meter. But we have a problem here, too. Radio waves come from wiggling electrons. So do light waves. But in our empty universe there are no electrons, so there's no light. We can't use light waves.
You might imagine that you will pace off the distance  one of your steps is about a meter. But there's nothing to stand on, nothing to push against. Furthermore, there's nothing to look at  you can't be sure if you're moving or just moon walking. On the Earth we know we're moving because when we look down the ground is moving past us, when we look to the side grass and bushes and trees are seemingly moving past us. In this empty universe there is nothing to look at, nothing to give us a reference point.
Let's imagine we brought a bucket of water with us into this empty universe. Now, let's start swinging the bucket of water around us in a big circle. We know what happens on Earth  the surface of the water takes a concave shape. The water surface is no longer flat. But why is this? How does the water know that it's moving in a big circle, and so it should change its surface to take on a concave shape? How does the water know it's not standing still and you're rotating around it? The answer, we believe, is that there is a big difference between these two ideas: if the bucket is standing still and you're rotating around the bucket, then when the bucket looks up at the distant stars it sees the stars as motionless. However, when the bucket is spinning around you the bucket sees the distant stars apparently all moving. The bucket has to believe either that it's moving, or the entire universe is spinning around it. We imagine that in either case the surface of the water will form a concave shape. We have to imagine this because we haven't figured out a way to spin the entire universe, so we have to guess.
However, when we spin our bucket in this otherwise empty universe there are no distant stars. The bucket cannot tell if it's spinning around you or if you're spinning around it. We imagine this means that in the empty universe, the surface of the water will stay flat no matter how you hold it or spin it. Again, we have no empty universe so we have to guess, but this is what we think.
In the end, we see that we can't measure distance. We can't speak of motion either. This empty universe has no concept of distance, no concept of motion, no concept of spinning or rotating.
Einstein worked out a complete theory of space and time and matter called General Relativity. General Relativity is expressed with an equation, G = 8 π T. 8 is just a number, and π is pi, the ratio of the diameter of a circle to the circumference, 3.14159. T is energy. In our normal day to day lives, T is just mass. This is why Sir Isaac Newton thought that gravity came from mass. But Einstein tells us that we have to include all forms of energy, which includes mass, heat, motion, pressure, chemical bonds, everything. In normal day to day existence all these other types of energy are very very small, and so mass alone is a very good approximation. Remember E = Mc^{2}. It's that c^{2}, the speed of light squared, that make all the other forms of energy look so tiny compared to mass.
L^{2} = X^{2} + Y^{2}
G is the instructions on how to make a metric, a formula for measuring distance. For example, on a piece of paper we measure distance the way Pythagoras taught us back in about 500 BC. The distance between two points is the square root of X^{2} + Y^{2}. Unfortunately, we don't live on a flat piece of paper. We live in a four dimensional space time. If we're very far away from all matter, the distance rule for our universe is the square root of T^{2}  ( X^{2} + Y^{2} + Z^{2} ). If we're near a large mass, like the earth or the sun, the distance rule gets more complicated. The distance rule near the Earth is square root of (1  2M/r)T^{2}  ( X^{2} + Y^{2} + Z^{2} ) / (1  2M/r) , where M is the mass of the earth and r is the radius of the earth in appropriate units. At the surface of the Earth, 2M / r is about one billionth. That means here on the surface of the Earth, a clock would be slow by about one second every 32 years compared to an identical clock which was far away from all galaxies. This is why Newton never noticed this: you have to be able to measure things to within about one part in ten million to notice this at all.
This very strange looking metric function is called the Schwarzchild Metric. You can get this metric by solving Einstein's equation of General Relativity, G = 8 π T. Interestingly, Einstein himself was not able to solve his own equation. A few months after he published his equation, a mathematician named Karl Schwarzchild solved it and found out that this is the metric function near a star or planet.
So in our empty universe, G = 8 π T, but T = 0. There's no mass, no energy, no pressure, nothing. In our empty universe G = 0. Einstein's equation tells us that we can't make a metric, we can't discuss the Pythagorean theorem, we can't have a yard stick. This is a mathematical way to say what we already figured out on our own. Or, we can make an even stronger statement: space and time are produced by matter. There is no stage unless there are actors too.
Now back to our experience in this empty universe. Let's ask about the other normal measurement, time. Can we measure time? Again, the answer is no. To measure time, we watch something changing and count the changes. For example, in a grandfather clock we have a pendulum that swings back and forth once each second. But our empty universe has no pendulums. More importantly, the pendulum only swings if there's nearby gravity pulling it downwards. In our empty universe there's no gravity, so even if we brought a pendulum, it would not swing. We might imagine a wristwatch  this is a little weighted disk that swings back and forth, driven by a little spring. The watch counts the swings on the little disk. But in our empty universe there's no wrist watch. We learned above that in our empty universe there's no concept of motion or rotation. If we brought a wristwatch the little disk would not move. In our universe the disk moves at the right speed because the disk has a carefully calculated mass. But in our empty universe, there's no mass, and there are no distant stars to give mass to things. There are also no distant stars to tell the wheel if it's moving or not. We might imagine using a digital watch. These watches have a little quartz crystal that is made to vibrate about a million times a second, and we count the vibrations. But in our empty universe, there are no quartz crystals. There's nothing changing, nothing we can count to measure time. So in our empty universe we can't speak of the passage of time.
Since nothing in our empty universe is changing, you won't be changing either. The vary process of thinking depends on changes in the electrical and chemical structure of your brain, but in the empty universe nothing changes so your brain won't either. Your thought processes will come to a complete stop in an empty universe. In fact, at this point we're a little confused about how God managed to say "let there be light" in a universe that was otherwise empty. That's ok, we don't presume to be able to understand everything about God.
In our universe time runs at a standard rate, but a bit slower nearby massive objects. We've calculated that time runs about a billionth slower at the surface of the Earth than it does far away from all mass. This is why gravity seems to pull things towards the earth. If we imagine a rock flying through space, when the rock passes by the earth it flies through the Earth's time field. In this time field, time runs just a bit slower for the part of the rock facing the Earth than for the part of the rock facing away from the earth. Since time is running slower on one side of the rock, that portion of the rock moves just a little bit slower. So the path of the rock seems to curve as it passes by the Earth. We used to say that the Earth's gravity pulled on the rock; now we see that the rock is simply trying to go in a straight line, but the straight line curves a bit due to the Earth's time field.
Let's look at this a bit more closely. Imagine you're standing on your front lawn playing catch with your kid. You toss the ball slow and a bit high to your kid, making it an easy catch. Your kid however, in exasperation, throws it back to you much faster on a much flatter arc. Einstein tells us that in both cases the ball is simply traveling in a straight line, and the straight line only looks curved to you because you're looking at it incorrectly. Let's see if we can understand this. Below is a picture of the two paths the ball takes. Both look quite curved to us, it's very hard at this instant to see how Einstein can claim these are straight lines.
Two paths taken by a baseball in a game of catch.
You and your son are standing 30 feet apart. In the first path, the higher arc, you throw the ball to your son at 30 feet per second, so it takes 1 second for the ball to travel from you to your son. Gravity at the earth's surface is 32 feet per second per second. Distance equals ½ a t^{2}. In the ½ second the ball is dropping, it falls 32 / 2*2*2 = 4 feet. So in the one second that the ball is in the air, it goes up 4 feet, down 4 feet, and across your lawn 30 feet. Your son throws the ball back twice as fast, at 60 feet per second, about 40 miles per hour. The ball is in the air for ½ second, and it's falling from its peak for ¼ second. The rise on the ball is 32 / 2*4*4 = 1 foot. So, the ball travels 1 foot up, 1 foot down, and across your lawn 30 feet. We're still having trouble seeing how Einstein can call these arcs "straight lines."
We're having a problem because we're looking at only two dimensions, distance and height. But we live in spacetime, we have to look at distance, height, and time. The speed of light is 1 foot per nano second, 1 billion feet per second. In the 1 second the ball takes to go from you to your son, 1 second of time goes by, which is 1 billion feet of time. In the ½ second the ball takes to go from your son to you, ½ second of time goes by, which is ½ billion feet of time. If this seems a bit confusing, it's because we're used to measuring distance in feet and time in seconds. This is like measuring the length of a horse in feet, but the height of the horse in hands. We know there's a simple conversion, three hands equals one foot. So, a horse that stands 15 hands tall is 5 feet tall. Similarly, we're accustomed to measuring time in seconds, but there's a simple conversion. The speed of light is 1 foot per nanosecond, so 1 foot equals 1 nanosecond, just as 3 hands equals 1 foot. One second equals one billion feet, which happens to also be three billion hands.
Now we will draw our baseball paths again, the same paths as above, but we'll draw them in 3d instead of 2d. The third dimension is time. It's pretty difficult to draw an arc that's 1 billion ticks long, so we'll just do our best and try to give an indication.
Two paths taken by a baseball in spacetime
Here I've tried to indicate how the paths look in spacetime. The ball goes up and down 1 foot or 4 feet; it goes left to right 30 feet; and it goes forward in time either ½ second or 1 second, which is equivalent to ½ billion feet or 1 billion feet. A line which is 1 billion feet long and 4 feet high in the center is actually a very straight line. Very roughly speaking, 1 billion feet is the distance to the moon, so a 4 foot high curve between here and the moon is pretty straight.
Why is this so hard to see? Why did it take 250 years, from Newton to Einstein, to understand this? The answer is that we caused our own confusion by thinking of time in seconds and space in feet. How did we come up with these crazy units, 5280 feet in a mile, 3 hands in a foot, 60 seconds in a minute? This all seems carefully designed to drive us nuts.
Originally, a foot was about the average length of a man's foot. A mile was 1000 paces of a roman centurian  apparently if you measured the distance between right hand root prints while the legions were marching, you would get 5.28 feet. Seconds are so old that we're not certain where they came from. Here's what I think: for thousands of years, man has been watching the stars. We noticed immediately that the same sorts of things happened each night, so timing them became interesting. The natural unit of time to a man without technology is his own heartbeat, which is typically a second. If you stand in one place and watch the sun or the moon set, you find it take 60 seconds from first contact with the horizon to being completely set. It's an incredible coincidence that from the Earth, the sun and the moon look to be almost exactly the same size  it's because of this coincidence that we have solar eclipses. So, I think it was natural to think that a minute  60 seconds  was an important amount of time. In another amazing coincidence, if you call 60 minutes an hour, then a day is precisely 24 hours. It could have just as easily been 23 ½ hours; in fact, the Earth is slowly slowing down, and in several million years a year will be 23 ½ hours. It just happened that humans started getting smart at a time when there were 60 heartbeats in a sunset, and 60 * 60 * 24 heartbeats in a day.
It's not to hard to believe that the fundamental laws of the universe don't have much to do with the average length of a man's foot, his heart rate, what the sun looks like from the Earth, or how far a Roman Centurian travels in a pace. In fact, later we'll see that there does seem to be a fundamental units of length in the universe. It seems to be about 4e35 meters, a terrifically small distance.
Let's think about horses for a minute. A horse's height is measured in hands, where a hand is four inches (don't ask me why, I've never owned a horse). The horse below stands about 15 hands high at the shoulders, and is about 7 feet long. When the horse rears up on its hind legs, if we were to measure the dumb way we might find that the horse is now 22 hands high but only 5 feet long.
A horse standing  The same horse rearing up 
The confusion here is because we're measuring height in hands, and length in feet. It would make a lot more sense if we were using the same units in both directions, like hands for both length and height. As it is, if we want to know the distance from the horse's rear hoof to his nose, we can't use Pythagoris' theorem, we can't say (15 hands)^{2} + (7 feet)^{2} = distance^{2}, because hands are not in the same units as feet. We could say something like (15 hands * 4 inches per hand / 12 inches per foot)^{2} + (7 feet)^{2} = distance ^{2 }. You can see this is a real pain  it's really not very convenient to use different units for different dimensions.
Similarly, we have a builtin confusion about space and time: we measure time in seconds and distance in feet or meters. However, knowing that the speed of light is the same for everyone, we can use the speed of light, c, to convert seconds into feet or meters. One foot of distance equals one nanosecond at the speed of light, so if we say a foot of time we mean the same thing as if we say a nanosecond. A meter of time is about 3 nanoseconds. From now on, we'll agree that we're always going to use the same units for time and space.
Velocity, distance per time, is now meters per meter, so velocity has no dimensions. The speed of light is now just 1 with no units. The speed limit on most freeways is 65 miles per hour. 65 miles per hour means 65 times 1000 paces of a roman centurian per 60 times the number of heartbeats in a sunset, which is also 60. You can see, these are pretty crazy units. 65 mph equals about c / 10,000,000. If physicists were running the highways, apparently highway signs would say "Speed limit 10^{7}." A traffic ticket for going 85 would read "excessive speed: 1.3*10^{7} in a 10^{7} zone." That's it, no units. If you were talking to God, I think He would tell you that the speed of light was one, and that 186,000 is some strange number that we humans made up.
Above we say that time was slowed down at the earth's surface by about 1 second in every billion seconds. Here we see that paths in the Earth's gravity curve by a couple feet in a billion feet. This is what curved spacetime means: at the surface of the Earth, the curvature is about one part in a billion.
An interesting question is, what is the radius of this curvature? A quarter has a radius of curvature of about ½ inch. The base boards in your house probably have a radius of curvature of about ¼ inch. A right hand turn on a residential street has a radius of curvature of about 30 feet. The surface of the Earth has a radius of curvature of about 4,000 miles, about 21 million feet. So how can we calculate the radius of curvature of spacetime at the surface of the Earth? Let's look at a portion of a circle for a second, and then do a small bit of math.
An arc of a circle 
height = h = r (1cos θ)
= r (1  (1  θ^{2}/2) ) = r θ^{2}/2 length = d = 2 r sin θ = 2 r θ r = d^{2} / 8 h 
In Figure 3, we see an arc of a circle. Actually this is an arc from an ellipse, but the difference is so tiny we'll use the easier math for a circle. The distance the ball is thrown is 2 * r sin θ. The height of the ball at midpoint is r * (1  cos θ ). We'll approximate sin θ as θ, and we'll approximate cos θ as 1  θ^{2 }/2. Now, the distance d we throw the ball is 2rθ, and the height h of the ball at midpoint is r θ^{2}/2. So, d^{2} / 8h = r. In our cases, the slow ball goes 1 billion feet of time forwards and 4 feet up, so r = 1 billion * 1 billion / 8*4. The fast ball goes forwards ½ billion feet of time and 1 foot up, so r = ½ billion * ½ billion / 8 = 1 billion * 1 billion / 32. It's the same radius of curvature, just as we would expect. How far is 1 billion * 1 billion / 32 feet? There are about 31,000,000 seconds in a year, and light travels 1 billion feet in a second, so a light year is 31 million * 1 billion feet. 31 million is about 1 billion / 32, so the radius of curvature of space time at the surface of the earth is just about 1 light year.
There's another way to figure this out. If you learned calculus, you learned that the acceleration is the inverse of the radius of curvature. The acceleration at the surface of the earth is 32 feet per second per second. The inverse of this is 1/32 seconds * seconds / feet. We need to convert this into just distance, so we'll multiply by the speed of light, 1 billion feet per second. Now, the acceleration is 1/32 billion seconds. We multiply by the speed of light again, and we get 1/32 billion billion feet. We just worked that out above as being 1 light year. So we can figure out the radius of curvature of spacetime at the Earth's surface by either knowing the acceleration of gravity or by throwing a ball and watching it.
The Earth has a very weak gravitational field. We can imagine a much stronger field. We could get a stronger field by either adding mass, or shrinking the size of the Earth so that we could get closer. If we look again at the metric function for a gravitational field, (1  2M/r)T^{2}  ( X^{2} + Y^{2} + Z^{2} ) / (1  2M/r), we notice something very strange. If r is ever the same as 2M, then the metric function says that time stops and distance is infinite. What does this mean?
We saw that a gravitational field slows down time. At the surface of the earth, gravity slows time by about 1 second every
billion seconds. Suppose the gravity field were a billion time stronger, because either the Earth weighed a billion times more or
it was shrunk to a billionth of its current size. It seems like time would slow in this case to a stop  our clocks would lose a
second every second. The radius of the Earth is about 3940 miles. There's 5280 feet in a mile, so that's about 21 million feet.
There's 12 inches in a foot, so that's about 250 million inches, about ¼ billion inches. So, if the Earth were shrunk to a ball
¼ inch in radius, about ½ inch across, then on the surface of the Earth time would stop. What does it mean to
shrink the Earth to the size of a marble? We have a name for this, in this case the Earth would be called a Black Hole.