My Summary of General Relativity

As you may know, Einstein’s theory of relativity is actually two related theories.  I wrote about the Theory of Special Relativity here, and that’s where most of the counter-intuitive weirdness is (time dilation, length contraction, and the unification of space and time into spacetime).  In contrast, General Relativity has only one big weirdness, the assertion that the “force” of gravity is an illusion, and that what we observe is just a manifestation of the fact that our spacetime is curved.  Hopefully you’ll be able to get your head around this by the end of the essay.  The math of general relativity is pretty advanced and hairy, but luckily it’s not necessary to understand this to get the gist of the theory, so I’m going to gloss over it.  It’s also not necessary to grok the concept of “spacetime” for our purposes here, so I’m only going to refer to “space”.

General Relativity gets its name from the fact that it describes all kinds of motion, while Special Relativity only describes the special case of inertial motion, which is motion that doesn’t include acceleration.  Einstein’s great insight about acceleration and gravity is that they’re the same thing. Therefore, General Relativity (GR) is a theory about accelerated motion and gravity.  

Before we get into all that, let’s review the theory that GR replaces, Newton’s Law of Universal Gravitation, which is usually given by this equation:

However, this equation is useless on its own.  To understand how gravity acts on massive bodies, we need to add Newton’s Second Law, which can be written this way:

You may be more familiar with the simple F=ma form, but since acceleration is just the change in velocity over time, the middle term should be understandable to anyone with a passing familiarity with calculus.  (For bonus points, understand that acceleration is the second derivative of position).  The reason to emphasize this is to show that the law of motion involving forces is a differential equation, which is an equation that relates quantities and their rates of change.  Einstein’s equations will also take the form of differential equations.  

The two equations above work together to tell us what to expect from gravity.  The first one says how to combine the masses of two objects with the distance between them and the mysterious constant G to calculate the force exerted on the body of interest.  We then plug the force into the second equation to calculate how the body of interest will move under the influence of this force.  

Simple and effective, Newton’s laws reigned for hundreds of years, and were utilized to calculate everything from the paths of celestial bodies and cannonballs to the interactions of billiard balls.  However, there were anomalies that Newton’s equations couldn’t explain.  In particular, the orbit of Mercury was slightly different from what was predicted by Newton’s laws.  The orbit precessed, which means the axis of rotation moved in a regular way, and nobody could explain this.

It took Einstein ten agonizing years to work his way from the Theory of Special Relativity to the Theory of General Relativity.  Reportedly, when he used his completed equations to calculate Mercury’s orbit and the precession was perfectly predicted, he was so excited he couldn’t sleep for days.

Einstein’s equations have the same two-part structure as Newton’s, but instead of a “force” being the currency traded between the two parts, Einstein’s equations use the curvature of space.  While curved space is a mental hurdle we’ll attack shortly, this is actually more intuitive, since it’s very hard to say exactly what a gravitational “force” is, beyond vacuous definitions like “the thing that makes masses move”.  Without further ado, here are the equations:

The first equation plays the role of Newton’s first equation, but instead of an “other mass” and a distance, the equation describes the curvature of space in terms of the matter density (expressed by the T term).  As I said, I’m not going to attempt to explain the math, but the quantities with the Greek indices are special matrices called tensors that contain factors for every dimension of space.  The second equation describes how objects move in a curved space.  This is clearly a differential equation, equivalent to F = ma, which is of course a special case solution when the space is flat and the second term vanishes.  In Newton’s universe, masses create forces and these forces make other masses move.  In Einstein’s universe, masses make space curve, and curved space makes masses move on paths called geodesics, which follow this curvature unless a force compels them otherwise.

Visualizing curved spaces is actually pretty easy if you limit yourself to two dimensions.  The surface of a sphere (such as the earth) is a great example you can intuitively explore.  If you compare a flat plane to the surface of a sphere, some pretty stark differences arise immediately.  On the plane, if you walk in a straight line forever, you never come back.  On a sphere, if you walk in a straight line you arrive back where you started in a finite time.  On a plane, parallel lines never meet.  On a sphere, parallel lines do meet (think of the meridian/longitude lines on earth meeting at both poles).  

Three-dimensional curved space is harder to visualize, and we’re not going to try.  It’s enough to accept that a creature trapped on a two dimensional surface who believes it is flat will be surprised at some of the phenomena of curved 2-D space described above.  If you can tentatively accept that you are a creature trapped in a curved 3-D space that you think is flat, you can anticipate that some surprising things will be observed if that space is actually curved.  

The main counter-intuitive result of the fact that we live in a curved 3-D space is that we have mistaken concepts of what “at rest” and “accelerating” mean.  We think we’re at rest when we’re standing, sitting, or lying down on something connected to the surface of the earth.  According to Einstein, this is not a rest state, it’s an accelerated state.  According to GR, the only time you’ve not been accelerating in your life is when you’ve been freely “falling” in earth’s gravity.  Probably the longest freefall you’ve experienced is the time between stepping off the high dive and hitting the water in the pool.

How can anyone say lying in bed is acceleration?  In classic Einstein style, we use thought experiments.  Imagine you’re in a closed box that appears to be an elevator car.  If this box was placed out in space an infinite distance away from any mass, you’d float freely, with no particular tendency to go to one side or another.  If you held a ball and then released it, the ball wouldn’t move.  Now imagine some force grabs one side of the elevator car and starts pulling at it, accelerating at 9.8 m/s^2.  Immediately, you would find one of the sides of the car rushing towards you.  As you hit the side, it would immediately seem to become the “floor”, and the side the force was pulling on would seem to become the “ceiling”.   If you regained your footing, it would feel just like standing up on earth.  If you held a ball in your hand, your hand would impart an acceleration to the ball which comes from the acceleration of the floor traveling through your body.  If you let go of the ball, it would stop accelerating and inertia would cause it to travel in the same direction at a constant speed.  Since the elevator car is accelerating, it would soon catch up to the ball and the “floor” would hit it.  Unless you knew what was happening, you would interpret this as the ball “falling” “down”.  This is how acceleration and gravity are the same thing, or at least indistinguishable.  

Back to you lying in bed, the matter comprising the earth and your house and bed are constantly accelerating and pushing on you.  This explains the force you feel the bed pressing against you, just like the ball experienced the force of your hand pressing against it in the elevator car.  If an empty shaft to the center of the earth suddenly appeared under your bed, you’d no longer feel the force of anything pressing against you.  Very relaxing and comfortable!  You might think that you’d start “falling” but this is only because you have a habit of comparing your motion to things on the surface of earth.  Relative to the curved space you inhabit, you’d finally be at rest (or in inertial motion, actually, so at rest in some suitable inertial reference frame), while objects on or in the earth accelerate away from you.

That’s really all there is.  It’s taken me quite a long time to become even remotely comfortable with the explanation above.   But if you want to understand how curved space explains gravity, changing your habitual notions of “at rest” and “accelerating” is the only way.

As remarkable as the Theory of General Relativity is, many mysteries about the nature of gravity remain.  Matter following geodesics through curved space makes some sense, and allows you to keep your intuitive idea of space as a set of locations that may or may not contain matter.  But the other half of the theory, where matter causes space to curve, doesn’t really have an intuitive explanation behind it.  What is matter and what is space?  How exactly does matter cause space to curve?  It may well be that space is just another mathematical fiction like Newton’s force.  It may be that there are particles called gravitons that transmit the force of gravity, and their discovery may make curved spacetime unnecessary.  We also haven’t united General Relativity and Quantum Mechanics into a theory of Quantum Gravity, so there will doubtless be many surprises when that happens.

My Summary of Special Relativity

As you may know, Einstein’s theory of relativity is actually two related theories.  The Theory of Special Relativity, published in 1905, describes the nature of space and time for the special case of inertial reference frames (coordinate systems that aren’t accelerating relative to each other and have no gravity).  The Theory of General Relativity, published ten agonizing years later, covers the more general case of accelerating reference frames, which Einstein asserted are the same as gravity.  The math involved in general relativity is much more advanced, but special relativity is where most of the counter-intuitive weirdness comes from.  This essay covers special relativity (SR).

A reference frame is a construct we use to describe a set of locations in the universe that are rigidly connected so they move together at the same speed.  Such constructs are useful because we can draw imaginary coordinate axes on them and thereby apply all the tools of mathematics to study events that happen in the frame.  Obviously, not everything in the universe is rigidly connected to everything else, and things move at different speeds.  Therefore, we’ll need many different reference frames to accurately describe the universe.  

Imagine you’re in the back of a pickup truck driving at a constant speed down a straight road, and your friend is standing on the sidewalk as you drive by.  As you and your friend wave to each other, you are in inertial reference frames that are moving relative to each other at a constant velocity, based on the speed and direction of the truck relative to the sidewalk.  When the relative velocity is constant, we say the motion of the reference frames is inertial (since the law of inertia says objects will travel in this way if no forces act on them).

It’s been know for hundreds of years that this kind of relative motion exposed deep laws of physics.  For example, if you reached out your hand and gave a high five to your friend on the sidewalk, both of you would feel the slap with equal force.  You could say your friend on the sidewalk is at rest while you and the truck are moving, but this is an arbitrary designation.  This can only be said if you affix the coordinate axes to the sidewalk.  If you fix the coordinate axes to the truck, then it’s you who’s at rest and your friend on the sidewalk who is moving relative to the coordinate axes or reference frame.  This thought experiment points out that the notion of absolute rest and motion that Newton championed is not as useful as the notion of relative rest and motion.  After all, both truck and sidewalk are on the rotating and orbiting Earth, which is in the solar system which orbits the galaxy, etc., but none of these other motions make any difference in how the participants in these experiments perceive each other.  

This simplest form of relativity is known as “Galilean relativity”, because Galileo Galilei was the first to state it clearly.  His most famous postulate of relativity is that if you were inside a windowless cabin on a ship on a clam lake, there is no experiment you could do that would tell you if the ship was at rest (relative to the lake) or moving in a straight line at a constant speed.  As soon as the ship accelerated or turned, you could feel it immediately.  We’ll come back to that when we talk about gravity, but for inertial reference frames, inertial (constant-speed straight-line) motion can’t be detected.

With Galilean relativity, velocities add up in simple ways.  This is immediately intuitive.  Imagine you’re in the back of the pickup truck with a paintball gun, preparing to prank your friend on the sidewalk.  If the truck is going 20 mph and the paintball gun shoots at 30 mph, then if your aim is true, your friend on the sidewalk will feel as if the paintball has hit him at 50 mph, because that’s its speed relative to the sidewalk.  The velocity of the paintball from the gun is added to the velocity of the truck and gun.  

This intuitive picture seemed to work, and Newton’s laws of mechanics are based on it.  The triumph of the theory of special relativity is to recognize that this isn’t quite right; it’s only an approximation.  At low speeds, it’s a very good approximation, so how was the inaccuracy discovered?  The answer involves the speed of light.

By the end of the 19th century, the speed of light was known to be finite and had been measured to some degree of accuracy.  The physicists of the day assumed that everything, including light, obeyed the principle of Galilean relativity.  Given this assumption, light had to be moving relative to something, and they gave this something the name “luminiferous ether”.  The ether was taken to be the substance which filled and defined a “universal rest” reference frame, relative to which everything else could be measured.  

Many experiments were done to try to measure the speed of the earth relative to the ether.  The first experiment that was done well enough to be trusted by the bulk of the physics community was by Albert Michelson and Edward Morely.  The Michelson-Morley experiment used an interferometer, the same instrument used in the recent LIGO apparatus that detected gravitational waves.  An interferometer uses light sent down two perpendicular arms that reflect it back to the right angle vertex.  This is generally done with a partially reflective mirror oriented at 45 degrees to a light source.  The mirror “splits” the light so that some is reflected and goes down one arm and some is transmitted and goes down the other arm.  When these partial beams are reflected back to the mirror, the two beams are partially reflected and partially transmitted again.  This combines light from both arms of the interferometer and sends the combined signal to a detector.  See this animation on Wikipedia (just the left hand side of the diagram for now).  

The lengths of the arms can be calibrated so the light that travelled on the two arms is in anti-phase when it reaches the detector.  This means all the crests of one light wave line up exactly with the troughs of the other light wave so they completely cancel each other out and the detector sees nothing.  After such a calibration, if a subsequent change in experimental conditions causes the light to take longer to travel one arm versus the other, then the light won’t be perfectly in anti-phase anymore, and the destructive interference will be incomplete.  Then the detector sees a signal, and the amount of signal can be used to measure the difference in time taken by the light that traveled the different arms.  

Michelson and Morley expected to be able to see a difference in the time taken by the two light beams when they changed the orientation of their interferometer.  In a sense, they were searching for the ether, the reference of absolute rest.  They expected that when one arm was oriented parallel to the direction of Earth’s motion relative to the ether, the path light would have to travel on this arm would be slightly longer than the light that took the path perpendicular to the apparatus’ motion relative to the ether.  See this analysis to understand why, but it’s just basic Euclidean geometry, algebra, and the assumption of Galilean relativity.  Or just look at the right side of the Wikipedia animation to get an intuitive feel for why the travel times were expected be different, resulting in a phase shift of the light beams.

The Michelson-Morley experiment is said to be the most famous and consequential “failed” experiment of all time.  Of course “failed” is the wrong word.  The experiment produced a “negative” result, meaning it didn’t show the expected signal from the expected path length differences of the two arms of the interferometer.  This negative result was a major anomaly demanding an explanation, since it contradicted the prevailing theory of a stationary ether and Galilean relativity.  And as all good anomalies do, this one spawned a scientific revolution.

If there truly was no way to measure motion relative to a fixed ether, this had immediate logical and mathematical consequences.  It was not Einstein, but several other physicists of the late 19th century who worked out these implications.  Hendrik Lorentz got a lot of the credit, and the transformations used in special relativity still bear his name.  Einstein’s major contribution was to show that the Lorentz transformations relating the velocities of relatively moving reference frames were a consequence of only two principles, and didn’t require the existence of an ether or a frame of absolute rest.  The two principles from which SR is derived are:

1. The laws of physics have the same form in all frames of reference.
2. The speed of light is the same for all observers in all frames of reference.

The first principle is an axiom, taken as given without proof.  Most people don’t have a lot of trouble accepting it, but the second principle is a bit more counter-intuitive.  It basically says that Galilean relativity is false.  It doesn’t have to be taken on faith, since that’s what you get if you believe the results of the Michelson-Morley experiment are correct.  They observed that light travelled the same speed no matter how they oriented their apparatus relative to the orbital motion of the earth.  People didn't notice the departure from Galilean relativity until the late 19th century because it only becomes significant for things traveling near the speed of light.  The value of the speed of light, c (approximately 3 x 10^8 m/s or 186,000 mi/s), also falls out of Maxwell’s Equations of electromagnetism, which led Maxwell to propose that light was an electromagnetic wave.

The Lorentz transformation is simply a set of formulas that allow you to convert distances and time in one reference frame to these same quantities with respect to another reference frame that’s moving inertially relative to the first.  When making up such formulas, it’s convenient to pick our coordinate systems so that the relative motion only occurs in the x-axis direction.  In that case, Galilean relativity has a simple transformation:

where x, y, and z are the coordinates in the “first” reference frame (which we’re choosing to call “at rest”), the primed coordinates are the “moving” reference frame, and v is the relative speed of the reference frames.  This is the transformation used in the example above to determine that the speed of the pickup truck and the paintball should be added together to determine what speed your friend on the sidewalk perceives (the minus sign could be a plus sign, depending on how you choose your coordinates).  

The Lorentz transformation is a little more complicated:

 where

The gamma factor is simpler than it looks.  If your speed is much less than c (the speed of light), then the second term under the radical () is very close to zero.  When that happens, gamma is very close to 1, and the Lorentz transformation approximately reduces to the Galilean transformation.  (This is why Galilean relativity seemed right for so long).  On the other hand, if your speed is very close to c, then the value of gamma skyrockets.  Here are some sample values:

The most surprising thing about the Lorentz transformation is that the x coordinate isn’t the only one affected by relative motion in the x direction.  The time coordinate is also affected.  This makes a little more sense when you realize the whole purpose of creating the transformation in the first place was to make the speed of light constant in all reference frames.  Since speed is distance/time, of course time must be affected in order to keep the distance/time ratio constant.

This mixing of space and time is what gives SR some of its truly weird and counterintuitive properties.  I don’t know if people can get used to it, but I haven’t yet.  If you plot the x and t coordinates (omitting y and z since they’re not interesting and we run out of dimensions to draw them in) along with the primed x and t coordinates, we get this remarkable plot:

Notice how the primed axes no longer appear to be at right angles to each other.  However, this is only because we’ve drawn the graph from the perspective of the unprimed coordinates.  Unexpectedly, if we adopted the other perspective, the primed axes would look “normal” and the unprimed axes would appear in exactly the same positions as the primed axes shown above.  This illustrates the counterintuitive reciprocal nature of the time and length dilation effects.  

For any “event” that occurs on this “spacetime” graph, the two different coordinate systems will ascribe different relative amounts of space vs. time to the event.  The rectangle and parallelogram on the diagram are meant to represent “dropping a perpendicular” to each coordinate axis in order to measure an event’s position and time.  Notice the different amounts of time and space ascribed by the two coordinate systems to the event at the upper right corner of the rectangle and parallelogram (compared to another imagined event at the shared origin).

The only things that are unchanged in both coordinate systems are the 45 degree x = t and x = -t lines.  These represent light rays travelling in the positive and negative x directions.  (The units have been chosen so that the speed of light is equal to 1).  Again, this shouldn’t be surprising since the whole purpose of all of this is to construct an analytic framework where the speed of light is constant in all inertial reference frames.  

Newtonian physics and Galilean relativity don’t rotate space and time into each other like this.  Newton never had the need to add a time axis to his motion diagrams because he assumed there was one absolute reference of time, and that it was possible to consider “all points in space at a given time”.  This is simply not possible when it’s a different time at different points in space that are moving relative to one another.  In SR, clock readings can only be made locally.  We normally make assumptions about what distant clocks read, but these assumptions are invalid.  As long as you insist on thinking of space and time separately, as if there existed “all of space at a given time” then SR will seem bizarre and paradoxical.  Well, there is no unique time for all of space.  Doesn’t exist.  You were misinformed.  Get over it.

The easiest way to start to accept that there is no absolute universal time is to understand the relativity of simultaneity.  That is, whether or not two events are simultaneous depends on how you’re moving relative to the events.  All other times reduce to this, since you’re always comparing one event of interest to another event (the tick of a clock).    If you believe the tilted coordinate axes described above, then this animation from Wikipedia should convince you that events can be simultaneous in one reference frame, non-simultaneous in another, and non-simultaneous in the opposite order in yet a third reference frame (the white bar represents the progression of time in the reference frames).  If you aren’t on board with the tilted axes, then try this video.  

How can events occur in different orders depending on who’s watching?  Doesn’t that potentially mix up the order of cause and effect?  No, because the order of events can only change for events with spacelike separation.   To understand what this means, you need to imagine that every event has a “light cone” that precedes and follows it.  The surface of the cone represents the possible paths light rays can take.  As shown below, this is a spacetime diagram where units are again chosen so the speed of light is 1 (it plots at 45 degrees), the vertical axis is time, and only two spatial dimensions are shown.

 

If one event is caused by another, then the effect event must be inside the light cone of the cause event, since some some signal or piece of matter must pass between them for them to be causally related, and nothing can travel faster than the speed of light.  Events with spacelike separation do not have light cones that overlap, so they are said to be causally disconnected.  For causally connected events, all observers will agree on the order of events, so causality is saved.  The diagram below shows several light cones that can be imagined to represent causally connected and causally disconnected events.  Only a timelike path can connect causally related events.

The ultimate effect of the Lorentz transformation is that relatively moving observers disagree about both time and distance.  This leads to the fact that moving observers experience both time dilation and length contraction.  Time dilation as derived from the Lorentz transformation is expressed with this formula relating the difference in time (𝝙t) between two events in primed and unprimed coordinate systems:

In this formula, the closer v gets to c, the smaller the quantity under the radical becomes, and the larger the primed 𝝙t becomes compared to the unprimed 𝝙t.  This means if you’re moving relative to a clock, the clock will appear to be running slowly, since your 𝝙t is larger than the 𝝙t in the clock’s reference frame.  However, in the clock’s reference frame, it appears to be running normally.   This result has been verified by flying atomic clocks around on planes and then later comparing them to ground-based clocks.  Taking this to the extreme, if you could actually travel at the speed of light, a clock at rest would appear to be standing still.  

The Lorentz transformation can be used to derive the following formula for length contraction of an object as seen from a reference frame moving relative to that object:

where L0 is the length of the object in the reference frame where it is at rest, v is the relative speed of the two reference frames, and 𝛾(v) is shown in expanded form on the right.  This formula shows that if the relative motion is much slower than the speed of light, the length is nearly the same in both reference frames.  As the relative speed approaches c, the length of the object as seen from the moving reference frame approaches zero.

Is there anything that stays the same in every inertial reference frame?  As a matter of fact, there is.  If you combine the space and time separation (deltas) between two events using the formula below, you get the quantity on the left hand side, which is called the “spacetime interval”:

The wonderful thing about the spacetime interval is that it’s invariant, meaning it won’t change when you view the two events from a different reference frame that’s moving inertially with respect to the first.  The two reference frames may see different amounts of space vs. time, but the spacetime interval will always be the same.

An even more remarkable takeaway from this formula is that c is essentially used as a conversion factor between time and space.  The remarkable thing is not that electromagnetic radiation is observed to move at this speed, but that such a fixed conversion factor exists at all.  

This conversion factor between time and space also leads to the last thing I’ll say in this essay on special relativity.  The fact is, everything, including you, is always travelling through spacetime at the speed of light.  If you are sitting still in your reference frame, then all of your spacetime travel is along your time axis, and you’re travelling on that axis at the speed of light, as the spacetime interval formula indicates.  As soon as you start travelling in space, then some of your velocity is used up by this travel, and your progress through time must be less than c (as seen from your former rest frame).  This is one way to look at time dilation (your clock will be seen as running slow by an observer you left behind).

What does this say about light itself?  The time dilation formula:

isn’t very helpful.  If you imagine a photon that left a distant star a billion years ago and entered your eye just now, then in your unprimed coordinates, 𝝙t is one billion years.  The reference frame of the photon is moving at c, so the denominator in the formula is zero.  This prevents us from calculating 𝝙t'.  No matter what 𝝙t is (no matter how far away the star is), the result is the same.

This says that light is expending all of its velocity on travel through space, so there’s no velocity left for travel through time.  This seems to indicate that time does not pass at all for a photon.  You could try to say that 𝝙t' for the photon is infinity, or zero, but neither one really works.  There is an incredible variety of answers to this question on the Internet, and most of them are pretty incoherent.  If you ever get a chance to talk to a photon and ask it what happened on its trip, please let me know.