The Curvature of Gravitation
(A project for 3Blue1Brown Summer of Math Exposition)
Motivation
The current theory of gravity tells a story about objects that curve (or bend or distort) space‑time. When no other forces exist, objects will move in straight paths at a constant speed. However, on a curved space‑time, such straight paths behave differently from what we expect. And then, the motion we observe in nature is of planets orbiting the sun, and apples accelerating towards the ground.
The aim of this presentation is to give an intuitive view of what is this curved space‑time.
We will explain
What is a straight line on a curved space, and how looking at straight lines can expose the curvature of the space.
How we can unite space and time into a single geometrical entity called a four‑dimensional space‑time
How gravity bends that space‑time, and how straight lines look on it
The visualization of gravity curving space should not have been difficult. Unfortunately, if you Google the term "curved space time" and go to the "Images" tab, you will be staring at various beautiful pictures with one common problem - they do not correctly describe the true nature of the curvature of gravity.
Quick History - Gravitation 1.0; Gravitation 2.0
Apples fall to the ground, and planets orbit around the sun. Isaac Newton connected both findings, explaining that bodies with mass attract other bodies, causing them change their motion. The calculation of the attraction force between mass 1 and mass 2 is given by the formula . F , equals G Mass 1 times Mass 2, divided by square of the distance between the masses
\[F=\frac{GM_1M_2}{R^2}\]
In 1905, Albert Einstein introduced the theory of relativity. He then tried to generalize the theory to include accelerated motion. The result (in 1915) was a more general theory, to be later known as "General Relativity". The theory gave a new explanation to both acceleration and also the attractive forces of gravity. So it became a new theory for gravity. The idea was that masses distort the "fabric of space‑time". In this space‑time, the motion of all the stuff (massive objects but also light!) is distorted as well, so they follow curved paths in the curved space‑time. A new formula gives the connection between "Energy-Momentum T", which is roughly the new name for Mass, and the geometric distortion of space‑time, now called "Einstein Curvature G". Both G and T are evaluated at each point of space, and at each moment of time. The formula looks simple, but this is misleading. \[G=T\] In reality, the equation is so complicated, that in most cases, we can't solve it (except for some super simple situations).
Let's forget about equations for a moment, and take a good look at what Einstein was suggesting.
Gravitation is NOT a force! Masses to not attract each other.
Gravitation is all about reshaping the geometry of the world, changing distances between points, and making new roads for other masses to travel on.
Then masses follow these new roads as their default path, without and sense of being forced to do so. The natural free motion is along the curved paths made by gravity.
These statements are a bit difficult to understand. To make it worse, Einstein did not suggest changing the geometry of our three‑dimensional space. We first need to include a new fourth dimension of time. Then, treat all that as a unified four‑dimensional object called space‑time. Only then can we start messing with its geometry.
What is this curved space‑time that makes gravity? How can we visualize it? This is the goal of our presentation. So let's talk about curved spaces.
Straight Lines on Curved Spaces
Let's compare a curved surface to a non-curved (or flat) one. We can tell the difference by just looking at them. It is less obvious from a "top view", so press "side view" for a better look. You can also use the mouse to change the view as you wish (Guide here). Can you see the curvature?
We can imagine drawing a straight line on a flat surface, but could there be a straight line on a curved surface? While there is an exact mathematical equation for a straight line on a curved space, we want to suggest a more intuitive way to look at it. Think about our surface as the field of a golf course, with all its hills and valleys. Then try to drive a golf cart on that field. Hold firmly to the steering wheel pointed at the forward position, and don't let it change. If you were driving on a flat field, your path would be straight. However, driving through hills and valleys will cause your cart to change direction. Even though your path is the straightest path possible.
Side View Top View
Let's look at examples of such paths. Press the following buttons to put some lines on both surfaces. Lines will start at the same point on each surface, and also with same initial direction, but the lines continue differently as they cling to the surface.
Two lines Parallels Spreading More Spreading Clear Lines
Some things that are special to straight lines on curved surfaces, like there could be more than a single straight line between the same two points, and line that start as parallel may meet. Such things do not depend on the view point, so they still exist even if we are viewing the surface from the top. From this view point the curvature is a bit obscured, but the straight lines continue to tell the tail of a curved surface.
This is important, because when we start talking about a curved three‑dimensional space, we do not have the option to step out and directly look the curvature from a higher dimensional point. So tracking the paths of the straight lines becomes a reasonable way to identify and estimate curvature.
While we live in a three‑dimensional world, understanding curvature on three‑dimensional spaces is not enough. The curved object, that drives gravity, has four dimensions. The three dimensions of space and another single dimension of time. Every "point" in that space is made out of 4 numbers (coordinates). Three numbers describe the location of the "point", while an additional number marks the time. The "point" is no longer a position, but rather an "event".
The Geometry of space‑time
Space Time
Fig #st1# - Space Time
We want to describe space‑time as a geometry. What could be the picture to draw? It should display all the events that happen, and for each event show the location and also the time. The picture should include all the events of some story.
For example, let's examine a race between a rabbit and a turtle. Both start at point A racing towards point B. The rabbit is faster, but once it got to the mid-point of the race, it fell asleep for some time, while the turtle continued past the rabbit. Later, the rabbit woke up and continued to run, but it was too late.
The whole story is shown in the above picture. The rabbit's path is in red, and turtle gets the blue path. Every point on this picture represents an event, that is something that happened at a certain exact location and at a certain exact time. The horizontal coordinate of an event holds the position, that is "Where", while the vertical holds the time, the "When". If you see that some participant in the scenario, has a vertical line, this means that while time is passing, position is not changing. This matches a participant staying put for some period of time, so no motion. On the other hand, we never see horizontal lines, as this would have meant a participant being at more than one position at the same time, which is not possible.
Such a map has two dimensions, one of space and one of time, jointly called space‑time. We see all the points on it (the events) and can draw lines between them. So we can treat them as if they belong to a geometry, using geometrical terms, like length, slope, etc. While it is two‑dimensional, the common way is to say that it has 1 + 1 (one plus one) dimensions. So we emphasize that there is one position dimension and one time dimension.
The physical space we live in is described using three coordinates to mark the position of a certain object. If we want to talk of our space‑time, it should have 3 + 1 (three plus one) dimensions. All together four dimensions.
Let's make a smaller step and talk about 2 + 1 dimensions. By that we mean, two dimensions of space with one dimension of time. Again, we meet our rabbit and turtle, but now they are free to move on a two‑dimensional plain. So they race again, but not from point A to point B, but rather on a circular track. The full space‑time will be, of course, a three‑dimensional thing (see below). The checkered floor marks the two coordinates of space, while the green arrow is the time coordinate, the "When".
Top view Side view Side 2 Run Back to Start
Do try playing around with the view of this 2 + 1 world. One interesting point of view is the "Top View". It shows all the places where the two buddies were passing through. And that is the circle or the racetrack. On the other hand, the "Side View" show the events including how time was playing into matter. So we can see each participant speed (this is the slope of the line, the more fast you are the less steep is your line). Convince yourself that you understand that the rabbit (red path) is generally faster than the turtle (blue path). See the vertical red line that represents staying in place.
Each point of view teaches us some part of the full story, so we need to always change our point of view, in order to get the whole story, and feel comfortable with what each part represents.
Another thing to look at is a single slice of time. Look at the current state of the floor. It represents the state of the world at a certain moment, the time of the start of the race. So we see the red and blue paths hit that floor plain, at a certain location. This is the starting point of the race. If we place the floor at a higher point, it would mean a later time into the race. If we continuously move the floor upwards, we would advance in time and could track both paths through the entire race.
Press "Run" to let time pass and track the points where the paths hit the floor.
Press "Back to Start" to bring the floor back to the starting time.
The Curved space‑time of Gravitation
So now we have some understanding about the thing we call space‑time. We have seen it from all angles, and understand the meaning of paths along space‑time. What is still missing is how gravity curve this space‑time.
One thing we can do, is to list the straight paths created by gravity. For this, we will use our knowledge about motions that gravity inspires. Yes, we are eventually back to "falling apples and orbiting planets".
So we will talk about the interaction between two bodies, one big against one small. Like the Earth against an apple, or the Sun against the Earth. In both cases, the motion of the big object will feel little effect, so it would be practically not moving (great! this is a simple vertical line in space‑time, standing still!). That big object will be the most significant contributor to the effect on space‑time curvature. So the smaller object will travel along a straight path inside this unchanging curved space‑time.
We will explore these paths below, where on the right there will be an animation of the motion, just like it looks in our three‑dimensional world, while the left is the space‑time view of a 2 + 1 dimensions. So pick a category from the tabs below. Then pick a path type to explore, read the description and press the activation button. When the path is seen, look at it from various points of view, and make sure you understand what's going on. And when more paths are added and the picture becomes cluttered, you can use "Clear Paths" below.
So pick a category and have fun!
Falling Apples
Orbiting Planets
Other Motions
Apples in free fall We know what happens to an apple when we hold it on our hand and let go. The apple moves straight down towards the earth with growing speed.
Click Fall to see the simulation of the motion on the right side, then look at the path in the space‑time on the left side. Did you notice that the initial direction of the apple in the space‑time view is vertical. This is because the apple was at rest at the beginning.
Throw the apple up in the air Another well known path of our apple is to throw it up un the air. We know what's going to happen. The apple will climb while loosing speed. Then from the top of the path, it will continue to accelerate down just as before.
Click Throw to see that simulation, and then see how this new path differs from the previous one, of just dropping the apple. Same starting point, but a different starting direction.
Throw the apple down to the ground In a similar manner, we can also ask what happens if we throw the apple down. Go ahead Throw Down.
All the three paths of the apple spread out of the same point.
Drop the apple, but at a later time Remember that we are looking for straight lines in space‑time experiencing gravitation. In that sense, the paths of an apple that is being dropped now is different from the path of making the drop a bit later or a bit earlier from now.
Click Drop Later/Earlier to see many similar paths, where the "let go" time was different.
The simplest orbit - circular path The exact circular orbit at a constant speed is pretty rare in nature, but it is the simplest orbit. The orbit of the Earth around the Sun, where the Sun stands in the middle of the circle, is an example of an orbit very close to circular.
Click Circular Orbit to see the simulation, then the path in space‑time. The path has a helix shape. Look at the path from the side, then rotate it around to convince yourself that the speed is constant (same slope everywhere).
Same circle opposite direction The circular orbit can go in both directions.
Click Other Way to see the opposite direction circle. Compare the space‑time paths of both circles. Where do they meet?
Elliptic orbits The most common shape of an orbit is the ellipse. The Sun is longer in the center, but rather at one of the two foci of the ellipse. The speed along the path is no longer constant. Speed is the highest near the Sun, and the slowest on the far edge (see if you can spot it in the space‑time path).
Click Elliptic to start the path.
Changing the starting point The circular path that you see, represents a lot of paths with a similar shape, but different starting point. In space‑time we can set the starting point (when time is zero) to be every point of the circle.
Click Circles to see a bunch of these circular paths, and notice the relation between them and the shape they all make.
Click Elliptic to see that same for the elliptic path. Look at it from Top View.
Hyperbolic path (not an orbit) These paths represent motions of objects that are not orbiting the center (the Sun). The motion is merely diverted a bit. The visiting object is coming from far away at a certain direction, and leaves at a different direction, influenced by the gravity of the sun.
Parabolic path (not an orbit) This path is very similar to the Hyperbolic path, but represents a case where the incoming object is coming with less energy, so it never leaves the gravity of the sun. However, the path does not make an orbit.
Side view Top view Clear All Paths
Final Words
Gravity warping space‑time is both easy and complicated at the same time. The guiding principle of going along the shortest path is so elegant and natural, however to see and understand the full entangled picture is close to impossible. Much like the Einstein's gravity equation \(G=T\).
I hope this presentation has contributed a little to clarify the picture.
Many things were not covered here
How does space‑time geometry look in cases where it is not constant?
What about more than one contributor to the curvature?
What would make a visualization of the full 3 + 1 dimensions?
What are Black Holes? and what has the speed of light to do with gravitation;