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| The twin paradox |
1. The laws of motion are the same in all inertial frames
2. The speed of light is the same in all inertial frames
The conventional view is that we live in a three-dimensional world where time proceeds at a constant rate for all of us independent of motion. But Einstein's theory shows that we instead live in a four-dimensional spacetime where space and time are interdependent. (Note: technical terms are explained in the Glossary at the end of this post.)
Lewis Carroll Epstein in his book Relativity Visualized presents a myth for intuitively understanding the special theory of relativity. As he puts it:
Why can't you travel faster than light? The reason you can't go faster than the speed of light is that you can't go slower. There is only one speed. Everything, including you, is always moving at the speed of light. How can you be moving if you are at rest in a chair? You are moving through time. (pp78-79)So, according to the myth, each of us are always travelling at the speed of light through 4D spacetime. Since Alice is always at rest with respect to herself, that sets her time direction. The three directions perpendicular to her time direction are the space directions that she perceives. If Alice changes her velocity, she rotates in spacetime such that she is now at rest with a new time direction.
When Bob is travelling at close to the speed of light, Alice measures Bob as aging slowly (time dilation) and shortened in the direction of motion (length contraction). This is simply the effect of Bob being rotated in spacetime relative to her and then projected onto Alice's coordinate system. But in Bob's (at rest) reference frame, his wristwatch is running normally and his shape and size are normal. This is analogous to how a round table appears oval when viewed from an angle. However the difference with relativity is that the projection is not simply perceptual, but is the outcome of collecting measurements from local observers who use clocks and rulers.
Alice's measurements of Bob from her rest frame can be represented by an Epstein spacetime diagram [1] as shown in Figure 1 below.
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| Figure 1: Bob is travelling at 0.6 times the speed of light in space (relative to Alice) |
Alice's proper time is on the vertical axis, while her space dimension is perpendicular to it on the horizontal axis (note: only the x-space dimension appears - the y-space and z-space dimensions are omitted). A radius represents 1 light year in a particular direction - vertical (for Alice at rest), horizontal (for a photon) or an intermediate direction (for Bob).
Alice travels 0 light years in space and 1 light year in time, i.e., she ages 1 year. At the other extreme, a photon travels 1 light year in space and 0 light years in time, relative to Alice.
According to Alice's measurements, Bob travels 0.6 light years in space. Bob's proper time can be calculated using the Pythagorean Theorem (where coordinate time is the length of the red arrow - Bob's worldline, Bob's proper time is the spacetime interval, and coordinate space is the distance Bob has traveled according to Alice's measurements): [1]
coordinate time2 = Bob's proper time2 + coordinate space2
Solving for Bob's proper time, where the coordinate time (the length of Bob's arrow) is 1:
Bob's proper time = √(coordinate time2 - coordinate space2)
= √(12 - 0.62)
= √(1 - 0.36)
= √(0.64)
= 0.8 years (or 9.6 months)
Since Bob has only traveled for 0.8 years at a velocity of 0.6c, the distance he has traveled in his rest frame is less than 0.6 light years. Bob's distance (according to his measurements) can be calculated using the usual distance formula:
Bob's proper space = velocity * Bob's proper time
= 0.6 * 0.8
= 0.48 light years
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| Figure 2: Alice is travelling at 0.6 times the speed of light in space (relative to Bob) |
In Figure 2, the same situation is shown, but from Bob's (at rest) reference frame. This is achieved by rotating the coordinate system such that Bob's arrow is on the proper time axis. Note that it is now Alice that has aged 0.8 years relative to Bob's 1 year.
These results may seem contradictory - how could it be that Bob has aged less than Alice in Alice's reference frame, but Alice has aged less than Bob in Bob's reference frame? The reason is that the time measurement that each person does of the other is a projection onto their proper time axis. To demonstrate this geometrically, Alice and Bob's reference frames are superimposed in Figure 3. Each observer is at rest, but measures the other moving obliquely. The symmetry is similar to two boats drifting apart from each other, and each perceiving the other as smaller. However, as noted earlier, the difference with relativity is that the projection is measurable, not merely perceptual.
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| Figure 3: Alice and Bob's reference frames superimposed - the situation is symmetrical |
If Alice and Bob meet up and compare notes, they will find no contradiction, i.e., they will be the same age, or just one will be younger than the other. To understand how this works, suppose that Bob returns to Alice at (very close to) the speed of light. His elapsed proper time will then be 0.8 years compared to Alice's now 1.6 years (since Alice ages a further 0.6 years during the return leg, while Bob does not age at all). Symmetrically, suppose that Alice returns to Bob at (very close to) the speed of light. Her elapsed proper time will then be 0.8 years compared to Bob's now 1.6 years. The opposite results in each case are due to the change in reference frames of the traveler (which is, in effect, an acceleration by the traveler that they themselves are able to detect).
These are instances of the famous twin paradox. An Epstein diagram can also show a more common rendering of the twin paradox, where Bob reverses course at the mid-point of his journey and returns to Alice at the same speed.
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| Figure 4: The twin paradox |
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| Figure 5: Alice measures the contracted length of Bob's rocket |
In Figure 5, Alice and Bob are part of the way along their respective journeys (note: the solid blue and red arrows are the same length), with the lengths of their respective rockets shown (exaggerated to be visible). Since Bob is traveling relative to Alice, Alice measures Bob's rocket to be shorter than its proper length, as projected on her spatial axis. For example, suppose the proper length of Bob's rocket is 0.1 light years. To calculate the length contraction, Bob's rocket can be rotated a quarter anticlockwise turn to lie along Bob's proper time axis (the red arrow) and projected onto Alice's proper time axis. Since the rocket is 0.1 times the length of the unit line, its projection against the proper time axis is 0.1 * 0.8 = 0.08. Rotating back, its projection against the space axis is also 0.08 light years. [2]
Bob would similarly measure Alice's rocket to be shorter than its proper length. (To see this, rotate the coordinate system such that Bob's arrow is on the proper time axis - then project Alice's rocket onto the spatial axis.)
In summary, both the time dilation and length contraction of an observed entity can be understood as projections onto the observer's proper time and spatial axes respectively. An entity observed by Alice to be travelling very close to the speed of light would be measured as having almost no elapsed time on its clock and almost entirely length contracted.
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[1] The most common spacetime diagram is the Minkowski diagram. However I've instead used the Epstein diagram here due to its intuitive representation (for a comparison, see Figure 6 below). An Epstein diagram is a space-proper-time diagram that shows the worldlines of objects traveling below the speed of light (i.e., on timelike, but not lightlike or spacelike paths). The basis for the Epstein diagram is explained below.
The Euclidean metric for calculating the total distance between points in 3 dimensional space is:
d2 = x2 + y2 + z2
This is just an application of the Pythagorean Theorem in a Euclidean space. Of interest here is that if a 1 meter ruler is rotated in space, the x, y and z coordinates can all change, but d will remain invariant.
The Minkowski metric for calculating the total "distance" (i.e., the spacetime interval L) between points in 3+1 dimensional spacetime is:
L2 = x2 + y2 + z2 - (ct)2
= d2 - (ct)2
It can also be expressed with an imaginary term:
L2 = x2 + y2 + z2 + (ict)2
= d2 + (ict)2
Or, as an alternative convention, the signs can be flipped (omitting the negative sign on L2):
L2 = (ct)2 - x2 - y2 - z2
= (ct)2 - d2
Minkowski space is non-Euclidean. Of interest here is that if a 1 meter ruler is rotated in spacetime, the x, y, z and t coordinates can all change, but L will remain invariant. Note that when d increases, so does t.
Rearranging the equation:
(ct)2 = L2 + d2
Which looks just like the Pythagorean Theorem! The spacetime interval L is a measure of the observed object's proper time (τ), d is a measure of the traveled distance, c is the speed of light (normalized to 1 in this post) and t is the elapsed coordinate time. Thus:
t2 = τ2 + d2
That is the basis for the Epstein diagram. In essence, Minkowski diagrams represent a hyperbolic equation (τ2 = t2 - d2) that is the difference of squares, whereas Epstein diagrams represent a circular equation (t2 = τ2 + d2) that is the sum of squares.
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| Figure 6: Comparison of Epstein and Minkowski spacetime diagrams |
Epstein diagrams are visually intuitive and quantitatively correct. For example, on the Epstein diagram, it is clear that Bob has aged less than Alice since his proper time is shown as less than Alice's proper time when they reunite. Also, the worldlines are the same length, representing the same elapsed coordinate time. Compare with the Minkowski diagram where Bob's shorter proper time is shown by a longer worldline than Alice's.
[2] The geometry is explained here. In particular, note the similar right-triangles indicated by the angle φ. These triangles can be rotated such that their hypotenuse lies along Bob's arrow, allowing the shorter side lengths to be projected onto Alice's axes.
Glossary
An inertial frame is a reference frame that is either at rest or moving at a constant velocity (i.e., not accelerating).
Proper time (or rest time) is the time τ as measured by a clock in a rest frame. On an Epstein diagram (see Figure 6), proper time is represented on the y-axis. On a Minkowski diagram, proper time is represented by the worldlines. While Bob's proper time is shorter than Alice's, it's shown as longer on a Minkowski diagram.
Proper space (proper length, space displacement, rest length, or rest space) is the distance d as measured using standard length rods in a rest frame. On both Epstein and Minkowski diagrams, the observer's proper space is represented on the x-axis.
Coordinate time (or observer's time) is the time t of a moving object measured by an observer at rest. Thus it is the same for all objects. On an Epstein diagram, coordinate time is represented by the worldlines (which are the same length on the diagram). On a Minkowski diagram, coordinate time is represented on the y-axis.
Coordinate space (or observer's length, coordinate length, or coordinate distance) is the distance d that a moving object travels as measured by an observer at rest. Thus it is the same for all objects. On both Epstein and Minkowski diagrams, coordinate space is represented on the x-axis.
Spacetime is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold.
The worldline of an object is the path that the object traces in spacetime.
The spacetime interval between two events is the length of the segment of the worldline connecting the two events. It is invariant, which means that everyone measures the same value.
Resources
- Relativity Visualized - book by Lewis Carroll Epstein
- In relativity, length contracts at high speeds. But what’s contracting? Is it distance or space or is there even a difference? - Ask a Mathematician
- Epstein explains Einstein
- Epstein Relativity Diagrams - Michael Weiss at Diagonal Argument
- Twin paradox flash player app (may require a flash plugin or an older browser)
- The Twin Paradox - Michael Weiss (Physics FAQ)
I have published a video using the interpretation of
ReplyDeleteEpstein to build Lorentz transformations. The operational advantages of this method are demonstrated.