Einstein’s Theory of General Relativity

It’s the year 2100. You wake up alone in a small, windowless room. The only other thing in the room is a small ball. Maybe the room is located in your city, but maybe it’s inside that new spaceship everyone’s talking about. How can you tell?

You pick up the ball and drop it. It falls vertically to your feet. You time the fall and calculate that the ball accelerates at 9.8 metres per second per second, exactly the acceleration of gravity at the surface of the Earth.

But a spaceship in the middle of deep space can also accelerate by that much, producing the exact same results. So where are you?

In 1911, Einstein formally proposed that gravitational mass (that which produces a gravitational field) and inertial mass (that which resists acceleration) were one in the same, and this became known as the “equivalence principle”. According to this principle, you can’t tell whether you’re in a gravitational field (such as on the surface of the Earth) or experiencing constant acceleration (a spaceship speeding up, pushing you to the floor, like the g-force of a roller-coaster).
 

Another example is the infamous “Vomit Comet”, officially the Weightless Wonder (see video below), used by NASA for training, and occasionally by Hollywood for filming. Just as with our example with the ball, there’s no way to tell the difference between free fall, and being in the absence of a gravitational field, say in deep space.

This principle led Einstein to consider incorporating gravity into the framework of his special theory of relativity, culminating in his General Theory of Relativity.

At face value, that doesn’t appear such a difficult thing to do. Until this point, the properties of objects in isolation could be described by equations with great accuracy. But what to do about gravity? How does one calculate the properties of a system in which acceleration can be due to either gravity or changes in velocity? It seems to depend on how you are looking at it.

That led to the idea of a “reference frame” – the stage on which the objects you are looking at play out their roles. There may of course be other frames in which the objects appear to behave differently, so we need a description of all the frames, and the way to relate them.

The trick was to consider space and time as a four-dimensional object in itself – not a fixed stage on which the objects are defined, but something that itself can change.

Space-time

 

Let’s say you and I are going to meet for coffee. How do you describe this “event”? One option is to look at a map – “I’ll meet you at the cafe on level two of the building that’s at G5 on the map”. We have described three coordinates: G, 5, and level two. This is another way of saying a set of x, y, and z coordinates. So that we both actually meet for coffee, we’ll also need to add a fourth coordinate: time – say 2:00pm. These four points are what we call a space-time event.

General Relativity says the map can be distorted; and our coordinates will depend on how that happens. If I were to bend the map a little, the distance between two locations changes.

If you measure and add the angles of a triangle on the flat map you would get 180 degrees. If you do this on the curved map, you get a little more or a little less (depending on which way it’s curved). In the same way, the universe itself can have areas of different curvature.

 Now for the mind-bending part …

 You might know that, in the absence of any forces, things like to travel in straight lines (thank you, Newton). What about when the space is curved? We can still talk about straight lines, but now the lines follow the curvature. Think about drawing a small, straight line on a basketball. You can draw a line all the way around the ball and arrive back at the starting point. It’s straight, but also curved.

Odd things happen in “curved space” that contradict what we expect from “flat space”. If you walk north ten kilometres, west ten kilometres then south ten kilometres, you would expect to end up ten kilometres west of where you started. Do that at the South Pole and you end up where you started! Technically this happens everywhere, but on a (non-cylindrical projection) map it’s obvious at the poles.

Now we can expand our definition and say objects not influenced by a force travel along straight lines in curved space. In particular, things with mass (or energy, thanks to E = mc²) follow these straight paths in curved space.

The experimental proof of this occurred during a solar eclipse in 1919 where starlight was observed to be bent by the sun. The amount of bending was predicted by Einstein, and not by the standard “Newtonian” theory.

So matter follows the curvature of space, but we know matter is the source of gravity, so the curvature responds to matter as well. In the words of American theoretical physicist John Archibald Wheeler, “Matter tells space-time how to curve, and curved space tells matter how to move.”

 What if we have lots of matter in one place? Imagine you are driving up a steep hill. There is some steepness that is too much for your car to manage, even at its fastest. In the same way, if we have a very large amount of matter in a very small area, the curvature becomes so strong that not even light (perhaps the fastest thing in the universe) is fast enough to get out. This is a black hole.

Beautiful curves

Starlight and black holes are fun, but what does this have to do with day-to-day life on Earth? Have you ever used the Global Positioning System (GPS)? It’s a common feature of mobile phones today, but it relies entirely on General Relativity to work.

We said our map could be curved so that the points in the space dimensions were closer together. Since space and time behave together as space-time, the same trick happens for time. If we have some large mass, the curvature in the time dimension means that the more curved the space-time is, the slower a clock ticks there (or appears to for someone in a less curved region).

There is a measurable difference between the rate at which your atomic clock ticks on the surface of the Earth, and that at which one in orbit ticks.

Without this correction, GPS satellites would not be able to tell you where you are with such accuracy.