Visualizing general relativity

Figure 1: The equivalence between gravity and acceleration

This post is part of a series on visualizing relativity.

As noted in the previous post, the first postulate of special relativity states that the laws of motion are the same in all inertial reference frames.

An inertial frame has no external forces acting on it, so either remains at rest or moves at a constant speed along a straight path. It would seem, then, that special relativity is applicable to gravity-free space but not to Earth where objects in free fall accelerate due to gravity.

This brings us to Einstein's thought experiment. Suppose you are in a compartment where you release an apple and observe it fall to the floor (see Figure 1). This occurs for a compartment at rest on Earth, due to gravity. But the same motion of the apple would also be observed if you were out in space and the compartment were accelerating upward at a rate of 9.8 m/s. There is an equivalence between these two scenarios - the laws of motion with respect to the apple are the same in both reference frames.

Now suppose you are in a compartment where you release an apple and observe it stay in the same position. This would occur for a compartment in free fall on Earth. The compartment and all the objects in it would be accelerating at the same rate, due to gravity though, relative to you in the compartment, the apple would be motionless. But the same (lack of) motion of the apple would also occur if you were out in space and the compartment were at rest. Again, there is an equivalence between these two scenarios - the laws of motion with respect to the apple are the same in both reference frames.

This is Einstein's equivalence principle which motivates the general theory of relativity. The laws of motion are the same for the compartment in free fall whether on Earth or in space. If this is so then it would seem that the compartment in free fall on Earth is an inertial reference frame. But, if so, then there can be no external force acting on it. That is, rather than accelerating, it is either at rest or moving at a constant speed along a straight path.

But how can this be? The apple certainly appears to be accelerating. Einstein's answer was that spacetime itself is curved and the apple follows a straight path (geodesic) through curved spacetime. This is illustrated in Figure 2 below (from this animation). In the Newtonian diagram, the apple is initially hanging on a branch. An upward force (green) is holding the apple and gravity (red) is pushing it down. When the apple falls off the branch, gravity pushes it down to the ground. In the Einsteinian diagram, only an upward force (green) is holding the apple which results in a curved trajectory through spacetime. When the apple falls there are no longer any external forces acting on it, so it follows a straight path through spacetime. Indeed, an accelerometer in free fall will measure zero since there are no forces acting on it.

Figure 2: Newtonian gravity vs Einsteinian curved spacetime

So that's the basic idea behind general relativity. A free falling frame is locally equivalent to an inertial frame and a gravitational field is locally equivalent to an acceleration. [1] We perceive the falling apple as accelerating downward but it's really us that are being accelerated upward by the Earth (as an accelerometer shows) and the falling apple's reference frame is inertial.

Let's now return to Alice and Bob from the previous post on special relativity and see how spacetime curvature affects one's predicted measurements of time and space. The following Epstein diagrams have been generated from Adam Trepczynski's relativity app.

Figure 3: Alice after 1 second in her rest frame

Suppose Alice (in her rocket) is motionless in space, far enough away from any massive objects so that there are virtually no gravitational effects. That is, local spacetime is flat.

As shown in Figure 3, Alice has traveled 0 light seconds [2] in space and aged by 1 second.

The yellow coloring indicates her rocket which, with respect to her rest frame, is not rotated in spacetime and is not contracted in length.

The clocks at the bottom indicate the elapsed coordinate time of 1 second (cyan) and Alice's elapsed proper time, or aging, also of 1 second (purple).

Figure 4: Bob after 1 second in Alice's rest frame

Figure 4 shows Bob (and his rocket) as measured by Alice in her rest frame. In 1 second of her proper time, Bob has traveled 0.6 light seconds in space and aged by 0.8 seconds (Bob's proper time).

The yellow coloring indicates Bob's rocket which is rotated in spacetime and is contracted to 0.8 of its rest length as measured by Alice.

The clocks at the bottom indicate the elapsed coordinate time of 1 second (cyan) and Bob's elapsed proper time of 0.8 seconds (purple).

Figure 5: Alice is stationary (in curved spacetime)

So far, I have described what Alice and Bob measure in situations that are unaffected by gravity.

Now suppose that Alice and Bob are on a massive planet that curves spacetime (note: the planet is to the right along the space axis, with the proper time axis bending away from it). Alice is on a rocket launch pad on the top floor of a (very tall) tower, while Bob is on a floor 0.6 light seconds below.

Note that Alice and Bob are being accelerated upward by the planet and tower. Since Alice is stationary (with respect to her spatial position on the planet), she moves only through time, aging by 1 second, as shown in Figure 5.

This is analogous to the apple's motion through spacetime while it is still hanging on the tree branch.

Figure 6: Bob is stationary (in curved spacetime)

Figure 6 shows Bob on the lower floor (i.e., to the right of Alice on the space axis). He ages 0.63 seconds as measured by Alice who has aged by 1 second, and he remains 0.6 light seconds lower in altitude than her.

This is an example of gravitational time dilation. Bob ages less than Alice since he is closer to the gravitating planet. [3]

Figure 7: Alice in free fall (in curved spacetime)

Now Bob catches the elevator to meet up with Alice on the top floor. Alice, in her rocket, launches off the top of the tower. She is initially at rest (with respect to the planet) and in free fall. Figure 7 shows that after 1 second of coordinate time, Alice has traveled 0.41 light seconds in space (downward) and aged 0.79 seconds.

That effect is what we call gravity. While Alice traveled in a straight line at a constant rate, the curvature of spacetime changed her direction from purely through time to partly through space as well. [4]

A way to visualize the diagram in Figure 7 is as a three dimensional truncated cone. The cyan line will no longer appear straight, but will increasingly curve down the cone.

This is analogous to the apple in free fall increasing in speed over time (relative to the tree branch).

Figure 8: Bob in free fall (in curved spacetime)

In Figure 8 Bob, not to be left behind, launches off the top of the tower with an initial downward velocity of 0.6 times the speed of light and in free fall. After 1 second of coordinate time, Bob has traveled 0.79 light seconds in space (downward) and aged 0.46 seconds.

As with Alice, even though Bob is traveling in a straight line at a constant rate, the curvature of spacetime changes his direction to be proportionally more through space than time than if spacetime were flat. Which is to say, Bob ages less and moves further through space than in Figure 4.

So that's the geometry of spacetime according to general relativity.

To review, the special theory of relativity states that the laws of motion are the same in all inertial (non-accelerating) reference frames. But what about accelerating reference frames? Einstein's subsequent insight was that acceleration and gravity are equivalent. Suppose Alice is on a train and experiences a jerk forwards as the result of the train braking. She can interpret this as the deceleration of the train. Or, equivalently, she can interpret herself as being permanently at rest while the world around her moves under the influence of a gravitational field. [5] Gravity, in turn, can be understood as curvature in spacetime. Thus acceleration can be understood as curvature in spacetime.

The consequence is that while any frame can serve as a rest frame (thus generalizing special relativity), the presence of spacetime curvature means that other frames may only be inertial with respect to that rest frame when in its local neighborhood and for a limited time. Keeping that in mind, the following principle can be stated:

General principle of relativity: The laws of motion are the same in all reference frames

So with reference to the general theory of relativity, what does this mean for Alice? It means that her frame can always be interpreted as a rest frame. And any frame that is accelerating with respect to Alice can be explained by the curved geometry of spacetime. Those rules apply equally for Bob, or any arbitrary reference frame.

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[1] The equivalence principle is a local equivalence - globally, gravity exhibits tidal effects. Consider also two free falling observers on opposite sides of the Earth. Both are in inertial frames yet, globally, they are accelerating towards each other. So measurements can only be compared locally where spacetime is approximately flat. Finally, consider that gravitation obeys the inverse square law and so will not be equivalent to uniform acceleration over large distances.

[2] 1 light second - the distance light travels in 1 second - is about 300,000 km (or 186,000 miles). The circumference of the Earth is about 40,000 km (or 25,000 miles), so light can travel around the Earth 7.5 times per second.

[3] For an analysis of the curvature of the time axis, see Gravitation and the Curvature of Space-Time. (See also Relativity Visualized, p145, and gravitational redshift.)

[4] Recall Epstein's myth from the previous post - Alice is moving at the speed of light through spacetime (as represented by coordinate time t). But as she progresses she is moving less in time and more in space, with the limit being the speed of light through space. That limit can't be reached in coordinate time t, but can be reached in a finite amount of proper time τ. (See also Relativity Visualized, p150.)

[5] My example is drawn from the following passage in Einstein's book, Relativity: The Special and General Theory:

'It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognises, in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a "real" acceleration (retardation) of the carriage. He might also interpret his experience thus: " My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under the influence of this field, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced."'

Einstein's thought experiment is similar to Galileo's in the sense that visual cues in the environment are discounted. While the frames can't be differentiated experimentally, an observer would normally differentiate them based on those visual cues (i.e., that the ship is moving on the water and that the train is decelerating). On this view, the observer's alternative interpretation in Einstein's thought experiment introduces a pseudo-gravitational field and does not involve actual spacetime curvature. General relativity is then reserved for real gravitational fields (due to matter) that exhibit as curvature in spacetime.

Glossary

The equivalence principle: The local effects of motion in a curved spacetime (gravitation) are indistinguishable from those of an accelerated observer in flat spacetime.

A geodesic is a straightest-possible line in a surface or a more general space. In the plane, the geodesics are straight lines, on the surface of a sphere they are great circles.

Gravitational time dilation: clocks in the vicinity of a mass or other source of gravity run more slowly than clocks which are farther away.

Resources

• Relativity Visualized - book by Lewis Carroll Epstein
• Curved spacetime - Epstein illustrations at UCLA Physics and Astronomy
• Epstein Diagrams for the General Theory of Relativity - Epstein explains Einstein
• Relativity flash player app (may require a flash plugin or an older browser)
• General relativity: Einstein vs. Newton
• Einstein Online Dictionary: inertial observer, equivalence principle, general theory of relativity
• The Twin Paradox: The Equivalence Principle Analysis - Michael Weiss (Physics FAQ)