Einstein’s most revolutionary idea is that gravity is geometry. The theorems of geometry are not eternal truths, but can be changed by mass and energy.

This is my first longer explanation of ideas that are well established in science, but not well understood by the public. To keep this from becoming too long and overwhelming, I have split the six parts of this explanation into three blog posts. (I&II, III&IV, V&VI)

The two parts for today are about how to write the Pythagorean Theorem for spacetime and how to modify it to include gravity.

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Prerequisites: My post Gravity is Geometry, Parts I & II and the Pythagorean Theorem. I am serious about explaining the key ideas of general relativity without using calculus or anything more difficult. If you find something hard to understand, please ask me for more clarification.

Originally Written: October 2016.

Confidence Level: Established science since the early 1900s.

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III. Spacetime

The first part of the theory of relativity is called special relativity. It involves objects and observers moving extremely fast. Although gravity does not make an appearance in special relativity, it is important to understand something about special relativity before trying to understand general relativity.

The way Einstein developed special relativity for the first time is interesting for historical reasons, but is not necessarily the easiest way to get intuition about the nature of reality. One particularly entertaining moment involved a thought experiment about what it would be like to surf on a light wave. Einstein eventually decided that this idea is even more outrageous than your current opinion of it. The conclusion of this thought experiment, as well as many other thought experiments and the actual experiment of Michelson and Morley, was that all observers measure the speed of light to be the same, regardless of how they are moving relative to one another. Nothing can travel faster than this universal speed limit.

Shortly after Einstein’s initial paper on special relativity: On the Electrodynamics of Moving Bodies, a mathematician by the name of Minkowski realized that special relativity could be understood by incorporating time into a new model of geometry.

To incorporate time into the Pythagorean theorem, you first have to multiply it by a speed: the speed of light, $c$. If you didn’t, you would find yourself trying to add miles to hours and that has never ended up well. Adding miles to hours $\times$ miles per hour works much better. That is not all that you have to do. Time also gets an extra negative sign in the Pythagorean theorem which distinguishes it from the spatial dimensions. $$ ds^2 = dx^2 + dy^2 + dz^2 – c^2 dt^2 $$

If $ds$ is zero, then $\sqrt{dx^2 + dy^2 + dz^2} = c \ dt$. The two events are separated in time by the time it takes for light to travel between the two spatial locations. This kind of interval is called light-like.

If $ds^2$ is less than zero, then the time part of the equation is bigger than the space part. This kind of interval is called time-like. Any event that could happen in our future or in our past would be separated from our current self by a time-like interval.

There is no need to get worried about $ds$ being an imaginary number, for two reasons. First, you only ever really need to use $ds^2$, which is always real. Second, reality may be stranger than you suppose, so it isn’t always a problem to get an imaginary number. One lesson that can be learned from this is that you have to measure time in a different way than you measure space. This lesson is obvious from everyday experience.

If $ds^2$ is greater than zero, then the space part of the equation is bigger than the time part. This kind of interval is called spacelike. Since nothing can travel faster than the speed of light, then you can’t in principle influence any event which is separated from you by a space-like interval. ^[1]Often we can’t influence events separated by us by a time-like interval either, but that is because of our own ineptitude; it is not guaranteed by physics.

Earth is very small compared to the speed of light. It only takes $0.1$ seconds for light to circumnavigate the earth. The spacetime interval between any two events on earth that occur more than $0.1$ seconds apart is time-like, regardless of how far apart they are geographically. It is much easier to see space-like intervals when the distance is larger. It takes light a few minutes to travel between Earth and Mars. When we send a rover to Mars, we cannot control the landing because the landing and mission control are separated by a space-like interval.

If you remember from above, the Pythagorean theorem is independent of rotations. A similar statement should hold for the Pythagorean theorem in spacetime.

A rotation which only involves the spatial dimensions leaves the Pythagorean theorem in spacetime unchanged. The spatial part looks exactly like the Pythagorean theorem in three dimensions.

There should also exist rotations which involve time as well as space. Since all rotations leave the spacetime interval unchanged, it is impossible to rotate a time-like interval into a space-like interval or either of them into a light-like interval.

It is possible to take any time-like interval ($ds^2 < 0$) and rotate it so that it is pointed along the time direction, i.e. so that $dx = dy = dz = 0$. It is also possible to take any space-like interval ($ds^2 > 0$) and rotate it so that it is pointed along a space direction, i.e. so that $dt = 0$.

Do these rotations involving time have any physical meaning? Yes, or I wouldn’t have mentioned them. A rotation which involves time is equivalent to changing from using a stationary observer to using an observer moving at some velocity. Let’s now apply this interpretation to the two rotations described in the proceeding paragraph.

Start with a time-like interval. The two events are separated in both space and time. If we choose an observer with the right velocity, it could pass through both events. To this observer, the two events are not separated spatially. She sees both events at the same location relative to itself, but at different times.

The second rotation contradicts our intuition and is a major source of confusion when first trying to understand relativity. Start with two events separated in both space and time that have a space-like interval. Since $dt \neq 0$, the two events are not simultaneous. However, if we can choose an observer moving at the right velocity, we could make $dt = 0$. This is the relativity of simultaneity. Two observers moving at different velocities with respect to each other could disagree on whether two events were simultaneous, without either of them being wrong.

If you want to learn special relativity in detail, there are many books and websites that you can learn from. In doing so, you will work through many thought experiments – and perhaps a few real experiments too – which unveil the counterintuitive results that occur when objects and observers are moving close to the speed of light. That is not my goal here. Instead, I am emphasizing that these results can be derived using a particular notion of geometry of spacetime.

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IV. Gravity

If you are a follower of popularized versions of science, then you have probably heard someone’s description of Einsteinian gravity. Often you will hear something like: Gravity bends the very fabric of spacetime. There is also the commonly used analogy of an object on a trampoline which changes the shape of the trampoline mat, deflecting the paths of other objects.

I am not particularly fond of these descriptions. Why does spacetime have fabric? The fabric or the trampoline mat sounds like something that permeates all of space and carries the force of gravity. This is an inaccurate description of gravity.

Instead, what we experience as gravity is the result of modifications to the laws of geometry.

To the classical physicist (anyone living before 1900), the laws of geometry are given. They are an eternal backdrop to the motion of objects. Gravity is a force by which the objects can influence each other. The force causes object to move, but does not change geometry.

Einstein’s greatest contribution to science was to show that the laws of geometry are not eternal truths. They change as a result of matter and its motion. Just like different objects influence each others’ motion, objects can also influence the laws of geometry. When changes in the laws of geometry influence the motion of the objects, we call that gravity.

What possessed Einstein to come up with such an outrageous theory?

We have already seen, in the above section on spacetime, that a moving observer can experience geometry differently from a stationary observer. Special relativity only deals with observers moving at a constant velocity with respect to each other, i.e. all observers are in inertial reference frames. It has long been known that if an observer is accelerating, that observer will experience extra ‘fictitious’ forces. These fictitious forces push you to the back or your seat when you step on the accelerator. They are also responsible for keeping you in your seat as you crest the loop of a roller-coaster.

All of these fictitious forces can be described geometrically. If you know how the observer is accelerating, then you can construct a geometry of spacetime in which there are no fictitious forces. Instead, the same motion happens as a result of the curvature of the geometry.

How is gravity connected to moving observers?

One of the most curious features of gravity is that it accelerates all objects the same way – if no other forces are present. All other forces act differently on different objects, depending on the object’s charge, or surface area, or some other property.

Fictitious forces are the only other forces that accelerate all objects the same way. Here, the explanation for this is clear. The objects themselves are not accelerating. The observer only thinks the objects are accelerating because of her own motion past the objects. Since the acceleration depends only on the motion of the observer and on nothing about the objects which appear to be accelerated, the acceleration due to fictitious forces is the same for all objects.

Gravity and fictitious forces share the key property that they accelerate all objects the same way. Locally, a gravitational field looks exactly the same as an accelerating reference frame.

The idea is nicely illustrated by a thought experiment. Suppose that you are inside of an elevator at the top of the Burj Khalifa. Suddenly, the cable holding the elevator snaps and you begin plummeting down the shaft. Two different effects are happening to you at the same time: you are experiencing the force of gravity and you are accelerating. Everything that you see inside of the elevator is also accelerating the same, so you are truly experiencing an accelerating reference frame. In this reference frame, everything falls the same. Since you, the floor and the ceiling are all falling at the same rate, there is nothing that pulls you preferentially towards the floor. The effect of the gravitational field is no longer apparent because you are accelerating with it. At least until air resistance in the elevator shaft becomes important or you hit the ground.

This example is not just a thought experiment. Something very similar happens in orbit. When astronauts are in orbit, they and their entire spaceship are always accelerating towards the ground together. They never hit the ground because they are going so fast parallel to the surface that the Earth curves away from them before they hit it. Since everything that they observe is accelerating the same way, they can no longer observe the results of gravity. Astronauts in orbit experience weightlessness because they are in a reference frame which accelerates with gravity, not because gravity is weaker in space.

This similarity between fictitious forces and gravity is called the equivalence principle. If accelerating reference frames are associated with modified geometries, then gravity should be as well.

Einstein’s equations for general relativity relate the laws of geometry in a curved spacetime to the presence of matter and energy. The motion of matter and energy through the spacetime is then influenced by the laws of geometry in this non-Euclidean / non-Minkowskian setting.

Einstein’s equations read: $$ G_{\mu \nu} = 4 \pi \, T_{\mu \nu} $$

I won’t go into all of the details of what this equation means because doing so requires a thorough understanding of differential geometry. The right hand side, $T$, is a measurement of the matter and energy at some location and how it is moving. The left hand side, $G$, is a (complicated) measurement of the Pythagorean Theorem at the same location. The meaning of this equation is thus: the Pythagorean Theorem is determined by matter and energy. If there is no matter or energy nearby, then the Pythagorean Theorem is the flat spacetime described in the previous part. However, if there is something nearby, the Pythagorean Theorem changes and describes a more complicated geometry instead. Matter tells spacetime how to curve. The second part of general relativity is a modification of Newton’s first law. Since straight lines often don’t exist in strange geometries, objects may not be able to move along straight lines. Instead, objects move along geodesics in the curved spacetime. Newton’s first law now states that objects move along geodesics through spacetime unless acted on by an outside force.

Gravity does not count as a force here. It instead determines the shapes of geodesics. The parabolic path traced by objects in free fall is not the result of a force which causes them to accelerate. The parabola is the shape of the geodesics near the surface of the Earth.

Next: Parts V & VI.