Warped Worlds

Aim / Overview

To explore, in an elementary and intuitive way, the ideas behind classical gravitation theory, otherwise known as Einstein’s general relativity.


  1. Protractor calibrated in “turns”. If none available, print the powerpoint slide below on overhead transparency film (laser printers will take this antiquated ware!)
  2. Ruler and pencil
  3. Small ball with dot drawn on it
  4. Drawing paper
  5. Three drawing pins
  6. Dark Coloured Thread
  7. 20cm Diameter Polystyrene Ball

Age Group

9 to 209

The Experiment

Simple Triangles

If the 360 degree turn angle definition is not yet wonted to the children, I recommend doing this activity in units of “fractions of a turn”: see my protractor below, you can print it onto a clear plastic sheet. Also, for this activity, the idea of an angle should be this: to find the angle between two lines:

  1. One imagines oneself standing on the plane containing the two lines and one’s feet are on the meeting of the two lines;
  2. One then turns so to face straight along one line;
  3. One then turns anticlockwise looking down on one’s feet (i.e. turn to the left) so as to face straight along the other line;
  4. The angle is the number of turns, expressed as a fraction of a turn one must make to change from facing one line’s direction to that of the other.

Have the children draw some triangles on flat paper, measure the angles at each corner and work out what the sum of these angles is. Do they notice a pattern? They may know the rule that the angles sum to 180degrees; for this exercise the rule changes to summing to half a turn.


Figure 1: A “Fraction-of-a-Turn” Protractor

Why the Rule?

(Optional section, depending on your children)

Now you may like to ask whether anybody can say why they add up to half a turn. This would be quite a deed for a child of nine, so you’ll almost certainly have to help them if you’re going to go down to this detail. This will depend on your children – be on the lookout for the sense of wonder slipping into a sense of overwhelmedness. Maybe you’ll simply want to leave it at the rule: angles in a triangle always add up to half a turn and not try to explain further. The central idea later on is that this angles-in-a-triangle-summing-to-a-half-turn rule is broken in special ways so that we need to understand that there is a rule in many situations in the first place. Many children will be happy with the idea that there is a rule after trying the measurement out for a few triangles.

However, if you want to give a little more detail, below is a sketch of something that may help.

Sum Of Angles

 Figure 2: Summing the Angles in a Triangle by Driving A Car

Imagine driving in the little green car anticlockwise around any triangle as shown. Think of the turns the car has to make at each of the triangle’s corner. The turn needed is a turn to the left and it is half a turn minus the angle at the corner. So when the car has gone right around the triangle and turned back to its beginning point, if we add up all the turns it has made it is three half turns minus the sum of all the angles. This is just the same idea that we can work out 96 + 99 + 95 as 100 – 4 + 100 – 1 + 100 – 5 = 300 – (4+1+5). It’s simply 300 less the sum of the shortfalls each of the numbers has from one hundred.

How else can we describe how much the car has turned? It mustr have turned a full turn because it is headed in the same direction as it was at the beginning of its trip. So:

Three halves of a turn minus the sum of all the angles is full turn.

Which number, when taken away from three halves is 1 (or, more simply) two halves?

Of course it is one half! So the angles always add up to half a turn, no matter what triangle the little car may go around.

Weird, Weird Spins

(Optional section, only done if the last “Why the Rule?” section has been done)

What if the triangle were not drawn on a flat surface? What then? In the above, we used the idea that we can find the total turn simply by adding all the turns together. Half a turn to the left followed by a quarter of a turn to the left is three quarters of a turn to the left and such like. This only works if we do our turns on one flat plane. Suppose that one of the corners of the triangle were halfway up a mountain, then the rule does not work. To understand why, you’ll need the help of a ball with a dot on it.

Composing Spins

Figure 3: Composing Spins: Socks on before Shoes is Different from Shoes on before Socks!

In the top sequence 1, 2, 3, beginning with the ball in position 1 we rotate about axis A, then about B.

In the bottom sequence 4, 5, 6, beginning with the ball in the same positon, we rotate about axis B, then about A.

The ball with the dot on it shows that turns add together nothing like numbers do: if you swap the rotations around, you’ll get a different result, whereas 2+3 = 3+2. This is altogether unlike what happens when all the turns are done in one plane as with the little car.

Although this may seem a bit weird, it really is quite normal in our World. Would putting your shoes on before your socks be different from putting your socks on before your shoes?

So we cannot add angles like we did with our little car above if the axis of the turn changes direction at all. Our reasoning does not work if the triangle is drawn on a curved surface. A curved surface means that the turns would be done about different axes.

Warped Triangles

Now we are going to meet the idea of a “straight line” on a curved surface. To best understand this idea, see the shortest distance flight paths calculated between Los Angeles and Stockholm, and between Melbourne and New York in the drawings below.

 LAX to Stockholm

Figure 4: Geodesic (Great Circle) from Los Angeles to Stockholm Projected to a Flat Surface Maps generated by the Great Circle Mapper – copyright © Karl L. Swartz.

Melbourne To New York

Figure 5: Geodesic (Great Circle) from Melbourne to New York Projected to a Flat Surface Maps generated by the Great Circle Mapper – copyright © Karl L. Swartz.

Drawn on flat maps they look bent but they are indeed the shortest paths one can go between the two points at either end of the path. Owing to the surface’s curvature, the idea of a straight line becomes a little weird, but the shortest line joining two points on a path is how we think about it now. On a curved surface, the shortest line along the surface joining two points is called a geodesic or a geodetic line. On a sphere, geodesics are circles and on the Earth they are all circles with the same diameter as the equator (roughly – the Earth is not quite a sphere but it’s not far off). Sometimes geodesics on spheres are called great circles.

So now we are going to look at some triangles on a curved surface. To do this, we stick three drawing pins into a polystyrene ball as in my photograph below, knot a thread around one pin, pull the thread tight around the triangular loop and tie it off back at the pin where the loop began. You’ll need to make sure that the loop is tight.

Geodesic on Foam Sphere

Figure 6: Making Geodesics on a Foam Sphere

By pulling the thread tight we make sure it follows the shortest path lines – the geodesics – between the drawing pins.

Now we measure all the angles at the corners in the triangle with a protractor as shown in my photograph below. One can do this much more accurately than you might think. The trick is to put the protractor’s centre at the drawing pin’s centre and make sure that the protractor’s plane is parallel to the tangent plane to the sphere at that point. Otherwise put, the protractor must be at right angles to the (imaginary) line linking the sphere’s centre and that of the drawing pin. When the protractor is aligned like this, if you look down on it “squarely” (i.e. your line of sight is also along this imaginary line and at right angles to the protractor’s plane), the great circles projected onto the protractor are indeed straight lines, so you can line the nought hair on the protractor precisely with one of the threads and read the angle of the neighbouring thread to within a degree or two (within a percent, if using the fraction of a turn protractor). Now we sum all the angles in the triangle.

What do the children notice about this sum?

Try doing this with different sized triangles. How does the sum of the angles depend on the triangle’s size?

You should notice that the triangle angles all sum to more than half a turn (180 degrees). The bigger the triangle, the bigger this sum.

Angle on a Sphere

Figure 7: Measuring the Angle on a Sphere

Think of the following example. On a globe of the Earth, choose two points on the equator. Now think about the triangle made from these two points and either the North or South pole.

This one should really get the message home when you ask what are the angles at the corners on the equator? They are of course a full quarter turn. You drop down to the equator from the North/South pole on a geodesic, you have to make a full quarter turn to follow the equator. The equator itself is a geodesic. So the angles add up to half a turn together with whatever the angle is at the North/South pole.

Afterthoughts and Discussion

Now we are going to do some imagining – some thought experiments. This might even be done on some mats as for our “How to Measure the Universe’s Size by Simply Looking into the Night Sky”.

We now imagine that we are little beings who live in a two dimensional World. There is no up nor down in this World. So it is very like being us but with no sky and no outer space. The third dimension doesn’t exist. You have to imagine very hard to do this. Imagine too that one island is like the whole Earth for us. Outer space is like the seas around the island, and this outer space is so unimaginably big that we don’t have too much idea about what lies out there or even how big it is. We can see distant “stars” across outer space, though.

For a long time people have thought that the angles in a triangle add up to half a turn. This is true for just about any triangle you can draw. But then one day, someone comes up with the question, “what if we could draw triangles far, far bigger than the Earth; would their angles still all add up to half a turn?”.

If the two dimensional World is curved, we can see that if the triangles are big enough, the answer may very well be no. Some curvatures don’t upset triangles like a sphere does. For instance, if one does the same experiments on a cone or a cylinder (it’s worth doing this with drawing pins on a cardboard tube, but the angles are harder to measure than for the sphere, so use  a BIG cardboard tube), then the angles still add up to half a turn. This is because the cone and cylinder can be rolled out into a flat sheet of paper. You can make a cylinder from a flat sheet of paper by rolling it up. The same is true for the paper chain links and Möbius strips we made in “Twist and Wonder”. They are all made by rolling flat paper so, in a weird kind of way, cylinders, cones and paper Möbius strips are all flat. They are made from flat pieces of paper that have simply been rolled up. Such rolling up does not change triangles drawn on their surfaces. Not so for the sphere and indeed any closed surface like the sphere wherein there are no edges.

The point about all this is that little beings who live on a two dimensional curved surface could find out that their World was curved without needing to go out of their two dimensions. You can tell the Earth is curved without going into space and looking down on the sphere but rather you can do it on the surface of the Earth simply by measuring the angles in big enough triangles drawn on the Earth’s surface.

So now we have to think very hard and ask what it would mean for our own three dimensional World to be curved! This is very hard to do, and you can’t really do it in your imagination. To understand how weird this is, if the Universe is not endlessly wide but closed, that is, if it doesn’t have any edges, then this means that straight line going off into space from the Earth’s surface can eventually come back to the Earth from the opposite direction, after it has gone around the whole Universe. It is likely that the Universe isn’t yet old enough for us to see light that has done around it several times, but some astronomers believe that they can see double views of some objects, one very early in that object’s life, the other at a much later time after the light from the first view has gone all the way around the Universe.

The great nineteenth century mathematician Carl Friedrich Gauss (1777 – 1855) took this idea very earnestly and actually tried to measure the angles of big triangles in space: these were not triangles along the surface of the Earth (which we know have angles that sum to more than half a turn), but rather lines of sight between the peaks of mountains. So Gauss’s triangles were ones through three dimensional space and his experiment was meant to show whether our three dimensional space were curved. He found that the angles of big triangles made like this indeed summed to half a turn, to within the accuracy of his instruments. So our three dimensional space, if it is “curved” must be very flat indeed.

Throughout the nineteenth century mathematicians (mostly, no one aside from mathematicians took the idea of curved space very seriously, certainly not physicists) grappled with this idea and it is safe to say that, by 1900, many of them had hit on the idea that the “stuff” in the World might have something to do with how our three dimensional World is curved. Experiments like Gauss’s showed the curviness must be tiny.

In the early years of the twentieth century, astronomers knew there was something wrong with their best theory for gravity – the one that Sir Isaac Newton (1642 – 1727) had put together. Newton’s theory let them calculate the orbits of planets and comets to astounding accuracy, but there were problems for things that go very near the Sun. In particular, Mercury’s orbit moved around a little (see the Wikipedia Page for “Apsidal Precession”) bit weirdly, as the drawing from Wikipedia below shows, and Newton’s theory simply could not explain this.

Mercury Orbit Precession

Figure 8: Precession of Mercury’s Orbit about our Sun

Newton’s theory showed that planets should orbit along ellipses (ovals) that stayed still, but the reality is that the ellipse itself slowly rotates with each new orbit as shown. Theoretically this happens for all planets, but it is an extremely small effect. It is most noticeable with Mercury, because Mercury is so near to the Sun.

After many years of thinking about problems like this, Albert Einstein had a hunch that gravity itself might be described by curved space and time. So, not only do we have to imagine our three dimensional space as being curved, one has to think of the Universe as a four dimensional object stretching out in space and time, and that this is the object whose curvature we need to think about. He wrote down a famous equation:

 $$\mathbf{G} =\mathbf{T}$$

On one side is a complicated mathematical object  related to the curvature of spacetime. But its essential idea is simple: it measures how much geometrical objects are distorted from their flat versions. For instance, by knowing  $\mathbf{G}$ a physicist or mathematician can tell you how much the sum of a triangle’s angles is different from the half turn rule we have in flat space. The other side of the equation is also complicated but simple in essence: it tells us how much “stuff” there is in the Universe and where. Actually $\mathbf{G}$ and $\mathbf{T}$ are like suitcases full of many numbers rather than only one number and there are complicated rules for unpacking the suitcase and telling what everything means. But the numbers in the  suitcase are big when there is a great deal of mass and or energy around, like at the centre of a star or inside the Earth. In empty space, all the numbers in $\mathbf{T}$ are nought and those in $\mathbf{G}$ tell us how gravity propagates through space exactly like a wave.

So, if $\mathbf{G}=\mathbf{T}$, then mass and energy tell spacetime how to bend, and, in turn, the bending of spacetime affects how mass and energy move. This description lets astronomers calculate how things in the Universe should move, and their calculations are found to be extremely accurate. If we do the calculations from Einstein’s equation for Mercury, we find that they tell us Mercury’s orbit should be exactly as it is seen to be by astronomers!

Another stunning way that we know Einstein’s theory is working is because the GPS on our phones and computers and in our cars works properly. The curvature of spacetime wrought by the Earth means that time actually runs every so slightly slower near the Earth than at the satellites. GPS devices calculate their positions taking the time distortion as foreseen by Einstein’s equation into account. If they did not, GPS would become more and more inaccurate: the errors of GPS devices  would increase by metres every few minutes. Within an hour or so, GPS would be utterly useless if Einstein’s equation were not right, or at least staggeringly accurate.

In Einstein’s theory, gravity is NOT a force. To describe how a satellite goes around the Earth, we simply calculate the geodesics through a space that is curved in the way Einstein’s equation says it should be. The satellite goes around the Earth because it is following the shortest path through space and time.

Likewise, how we feel heavy is described a little differently. We feel our bottoms pushed on our seat because the it so happens that the shortest paths through space and time are not the paths that say we should sit still at one point on the Earth, the shortest paths are actually going through our feet accelerating towards the Earth’s centre. Now, an object must follow its geodesic through spacetime if there is no force on it, and, if it is not following a geodesic, there must be a force on it. We can’t follow geodesics through spacetime because our bodies can’t pass through the ground. This last fact has nothing to do with Einstein’s theory – it comes from other parts of physics. But, given that the ground does hold us back from following our spacetime geodesic, it follows that the ground must be exerting a force on us. We feel heavy because our seat or the ground is constantly thrusting upward on us to stop us following the spacetime geodesics that Einstein’s equation would foretell. We feel our seats holding us back from accelerating towards the Earth’s centre.