Chapter 3: Gravity and Orbits

General Relativity

Critical Questions:

  • What is General Relativity?
  • How do we know that general relativity is correct?

Ok, if you’ve already read the previous post, we can now dive into the awesomeness of General Relativity.

I’ve already said that Isaac Newton was bothered by his own theory of gravity because it seemed to involve things affecting each other through empty space, without ever coming into contact. In fact, nobody felt comfortable with the idea of a force that acted at a distance: it seemed too much like fantasy and not enough like solid science.

Isaac Newton

It's probably what he was frowning about in this portrait.

The only problem was that this theory worked very, very well. It explained almost everything related to gravity, most notably the motions of all of the planets and stars in space. It was even used to successfully predict the existence of the planet Neptune based on the motions of Uranus (albeit almost 200 years later), and verified predictions are the true test of any scientific theory. But Newton wrote the following in a letter six years after publishing Principia:

“That gravity should be innate, inherent and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me a great absurdity, and I believe that no man who has in philosophical matters a competent faculty of thinking, can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial I have left to the consideration of my reader.”

This was a daunting challenge, and one which most people judiciously ignored until Albert Einstein began working on General Relativity in 1907.

According to Einstein himself, there was one particular thought which led him to begin work on GR, and it was that a person in freefall experienced weightlessness, as we’ve already discussed. Another way to approach this idea is through a related thought experiment he came up with later. Imagine that you wake up one morning in a box about the size of an elevator. You have no idea how you got there, the doors won’t open, and there are no windows to the outside world. Looking around, you notice a small bronze plaque on one of the walls. It reads:

This room is either:
(A) stationary, on the ground, on Earth, or
(B) in outer space and accelerating upwards at 9.8 m/s2.
Which one?

Hanging just below the plaque are a stopwatch and a meter stick. You decide to accept the plaque’s challenge. You grab a pen from your pocket and make a series of measurements, eventually deducing that that whenever the pen is dropped, it accelerates downwards at 9.8 m/s2. Therefore, you decide smugly, the room must be accelerating upwards through space, leaving the pen to follow Newton’s First Law (moving at a constant speed) and “fall”, relative to you, at the same rate the room is accelerating up.

But suddenly, and with some dismay, you remember that everything in freefall on the Earth also accelerates downwards at 9.8 m/s2.

So aside from the problem of being stuck in a small room, you also have no way at all to scientifically determine whether you’re in situation A or situation B. Both scenarios are effectively equal, from a physics point of view.

Einstein was fascinated by what these two examples implied, which is that acceleration and gravity are, in some sense, functionally equivalent. From there, he spent eight years developing the startling theory now known as General Relativity. Unfortunately for us, this is the point where we’ll have to make a frustrating leap across the difficult mathematics to the final result. We may not see exactly how GR solves the elevator problem, but we can trust that Einstein, at least, knew what he was doing.

Einstein on a bike

He did.

The central idea of GR is that space and time are not separate from each other but are bound together as ‘spacetime’, and that the presence of matter causes spacetime to curve. This is one of those ideas where, if anyone tells you they really understand it, they’re probably lying. But let’s look at each of these two statements separately.

Before Einsteinian relativity, you could trust time and space to behave normally no matter what the other one was doing – checking your watch on a speeding airplane, for example, would be just like checking your watch while sitting at home. But Einstein’s theories showed that moving through either time or space affects the way they behave. According to Special Relativity (which we’ll discuss later, remember), different kinds of motion affect your measurements: moving more quickly through space means moving more slowly through time, for example. Thus, we can no longer talk about just space or just time. It’s now all spacetime.

That stuff will make more sense after you’ve heard about Special Relativity. But the part of GR we can spend more time with now is spacetime curvature.

Imagine all of the empty space around you as being filled with an invisible, 3-dimensional grid. Before GR, it was assumed that the lines of this grid were straight and that space obeyed the rules of Euclidean geometry. In GR, those lines are thought to bend or curve when any matter is nearby.

A common analogy for this involves a bedsheet. Imagine holding a bedsheet taut by stretching out all four corners; the sheet now represents empty space. Now put something onto the middle of the sheet, like a baseball, while still holding it taut – if you’ve picked the right size of object, the sheet will bend down around the ball, but will remain mostly flat elsewhere.

Spacetime curvature

It's exactly like this, except it's completely different.

In some ways, this is a good analogy for GR: anything with mass will bend the lines of space. And this bending is what actually causes gravitational attraction – it turns out that mathematically, this curvature can explain gravity without having to talk about forces at all.

However, there are a number of reasons why this idea is extremely difficult to wrap your head around. First of all, the sheet is a 2-dimensional surface, while space is 3D. When the sheet bends, it bends into the third dimension; where does space bend into? The answer is that these are two very different types of bending – mathematically speaking, you might say that space curvature is more about changing the definition of a straight line.

The second problem with this analogy is that it’s not just space that bends, but spacetime. Time curvature has some rather interesting side effects, as we’ll see shortly, but we can’t picture it as part of the grid because humans are incapable of imagining things in four dimensions.

But the real reason this is so tough is that we’re once again using words – an imperfect and imprecise language that was really not built for this kind of task – to describe an abstract mathematical concept.

So if, for now, you’ll accept that mass bends spacetime, let’s talk about some of the consequences of GR.

One of the most interesting things that GR predicted was that light beams would bend when passing near massive objects.[1] According to classical physics, light always travels in straight lines; in GR, these lines are still “straight” (in some sense), yet they seem bent because they are moving through curved space. The result is something like the diagram below:

Light deflecting around a massive object

Thanks to his work with massive objects, Einstein was also the regional Your Mom Joke champion for seven consecutive years.

Light deflection was the first experimentally verified prediction made by GR, and was thus responsible for convincing the world of its accuracy and for making Einstein a household name. In 1919, Sir Arthur Stanley Eddington led an expedition to an island off Africa’s west coast to observe a total solar eclipse. Photographic images of the sky showed that a number of stars were not where they should have been – their light had been deflected by the mass of the sun – and these differences matched perfectly with the mathematics of GR.

The combined effect of all of this bending light is called gravitational lensing. A very massive object, like a galaxy or a black hole, will cause distortions around it, like looking through a glass lens. Gravitational lensing can also produce multiple images of the same star: if the light from one star bends around both sides of a massive object, it seems to be coming from two sources. This is a perfect example of non-Euclidean geometry: two beams of light diverge, travel in straight lines, and then meet up again – something that could never happen if spacetime wasn’t curved.

Galaxy Cluster Abell 1689, taken by the Hubble Space Telescope in 2002

If you enlarge this image (click it), you can clearly see the lensing effects of a massive galaxy cluster called Abell 1689.

Another prediction of GR is that time slows down in a gravitational field: the stronger gravity is, the slower time passes. And I don’t mean that the clock in your cellphone won’t work correctly, I mean that time itself actually passes more slowly wherever gravity is stronger. We’ll see more examples of this “time dilation” in the section on Special Relativity, but it too has been verified experimentally many times over. GPS satellites need to use the equations of general relativity to correctly track time at their high altitudes, where the strength of gravity is much weaker than on Earth.

By now, GR has made so many correct predictions that it has been accepted as the current theory of gravity. Its biggest flaw is that nobody has yet been able to reconcile it with the equally successful theory of quantum mechanics, which is excellent at explaining particle behaviour. But in the meantime, it’s also given us the fantastic idea of black holes, which we’ll finish this chapter with.

Big Ideas:

  • General Relativity says that space and time are bound together as spacetime, and that objects with mass curve the spacetime around them.
  • The curvature of spacetime explains how gravitational interactions happen.
  • One of the consequences of spacetime curvature is the deflection of beams of light, or gravitational lensing.
  1. I’m using ‘massive’ in the technical sense here, meaning ‘having mass’, as opposed to the common definition of ‘very large’.
3.6 General Relativity   (from Chapter 3: Gravity and Orbits)



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