The Relativity of Simultaneity

The relativity of simultaneity is a consequence of Einstein's special theory of relativity.

In the Newtonian conception of space and time, time is both absolute and independent of space. In Newtonian physics, for example, whether two events happen at the same time is an objective question, completely independent of your point of view: They do or they don't, and that's the end of the matter.

Special relativity tells us the Newtonian conception of time isn't quite right. According to special relativity, time is neither absolute nor fully independent of space. That linkage between time and space is why, when we change our point of view via the Lorentz transformation known as a Lorentz boost, we must transform our time coordinates as well as our spatial coordinates.

One consequence of relativity's merger of time and space into a unified spacetime is that simultaneity is not absolute. If two distinct events are simultaneous in one coordinate system (of the special type known as an inertial reference frame), then there are equally valid coordinate systems of that type in which those events do not occur at the same time. Whether those events are simultaneous is therefore not an objective question. Its answer depends upon the coordinate system you prefer to use.


Outline of this essay

1. Precise statements of the relativity of simultaneity.
2. Proof that the relativity of simultaneity is a consequence of basic principles of special relativity.
3. Empirical evidence for the relativity of simultaneity.
4. Question:
   What do "observers" have to do with the relativity of simultaneity?
   
   Answer: Hardly anything.
   
   The more interesting question is why so many expositions of the relativity of simultaneity mention observers. The answer to that more interesting question is mostly historical.
5. Question:
   Is the relativity of simultaneity a perceptual phenomenon?
   
   Answer: No.
   
   The more interesting questions are

   • Why do some people make the mistake of thinking it's just a perceptual phenomenon?
     
     Partly because some people overlook the difference between "subjective" and "perceptual", but there's a little more to it than that.
   • Why do a few expositions of the relativity of simultaneity mention perception?
     
     A very few expositions mention perception for some good reason, but sloppy exposition is the more common reason.

I will use the phrase "Recall that...", together with hyperlinks, to state some basic principles and definitions of special relativity. Readers who are unfamiliar with those principles should follow the links.

Recall that special relativity applies only to flat (Minkowski) spacetime, which I will assume throughout this essay. (The relativity of simultaneity becomes more general in general relativity, but that generalization involves substantially more advanced mathematics. The basic ideas of special relativity are accessible to anyone with a solid grasp of high school algebra and an interest in physics.)

For the reasons given in Part 3, I will assume special relativity is a correct theory. My goal here is to explain how the relativity of simultaneity follows from that theory.


Part 1: Precise statements of the relativity of simultaneity

Recall that a Cartesian coordinate system for 3-dimensional Euclidean space assigns numbers (coordinates) x, y, and z to every point of the space in such a way that the distance ds between two points of the space satisfies the equation

ds² = dx² + dy² + dz²

where dx is the difference between the x coordinates of the points, dy the difference between the y coordinates, and dz the difference between the z coordinates. That equation is of course the Pythagorean theorem generalized to three dimensions. Taking the positive square root of both sides of that equation defines the Euclidean metric (also known as the Euclidean distance function).

Recall that, in special relativity, an inertial reference frame is a 4-dimensional coordinate system for Minkowski spacetime that assigns numbers t, x, y, and z to every point of the space in such a way that the spacetime interval between two points is given by the equation

ds² = − c² dt² + dx² + dy² + dz²

where dt is the difference between the t coordinates of the points, dx the difference between the x coordinates, dy the difference between the y coordinates, and dz the difference between the z coordinates. That equation is also known as the Minkowski pseudo-metric or (by a minor abuse of terminology) the Minkowski metric. An inertial reference frame is said to be inertial because the straight line obtained by interpolating between the coordinates of two distinct points represents a path through spacetime of some particle moving at constant velocity, under its own inertia, hence free from acceleration.

Recall that the signs of the terms on the right hand side of that equation form a metric signature. The spoiler explains my preference for the (− + + +) signature I'm using throughout this essay.

Quote:

Originally Posted by Wikipedia In general, but with several exceptions, mathematicians and general relativists prefer spacelike vectors to yield a positive sign, (− + + +), while particle physicists tend to prefer timelike vectors to yield a positive sign, (+ − − −).

The relativity of simultaneity doesn't have much to do with particle physics. The (− + + +) signature is more common in books on relativity, such as Misner, Thorne, and Wheeler.

Recall that, with the metric signature I'm using (which is the opposite of the metric signature used by this Wikipedia page), a negative value of the spacetime interval is said to be timelike, a positive spacetime interval is said to be spacelike, and a zero value is lightlike.

• If the spacetime interval between two events is timelike, then the straight line interpolated between the two events represents a path that could be followed by some unaccelerated massive particle traveling at less than the speed of light.
• If the spacetime interval between two events is spacelike, then the straight line interpolated between the two events represents a path that no particle could follow without exceeding the speed of light, which (according to the theory of relativity) is impossible.
• If the spacetime interval between two events is lightlike, then the straight-line path between the events could only be followed by a massless particle traveling at the speed of light.

Here are a couple of facts that can be proved using essentially the same technique I'll use to prove the relativity of simultaneity. I'm stating them here because some readers will be reassured by the absoluteness of the first fact, while the second fact states a relativity of spatial co-location analogous to the relativity of simultaneity.

• If the spacetime interval between two events is timelike, then one of the events happens before the other, and all inertial reference frames agree on which event happens first.
• If the spacetime interval between two events is timelike, then there are inertial reference frames in which both events have the same spatial coordinates but different time coordinates, and there are also inertial reference frames in which the events have different spatial coordinates as well as different time coordinates.

Generalizing those facts to general relativity is complicated by the fact that general relativity is consistent with causal loops. (There are no causal loops in the flat Minkowski spacetime of special relativity.)

With all that background out of the way, we can now give two precise statements of the relativity of simultaneity.


Relativity of Simultaneity (Version A)

If E₁ and E₂ are distinct events that occur at the same time in some inertial reference frame R, then there is another inertial reference frame R' in which E₁ and E₂ occur at different times.

Relativity of Simultaneity (Version B)

If events E₁ and E₂ are separated by a spacelike interval, then there is an inertial reference frame R₁ in which E₁ occurs before E₂, an inertial reference frame R₂ in which E₂ occurs before E₁, and an inertial reference frame R₀ in which E₁ and E₂ occur at the same time.

Part 2: Proof that the relativity of simultaneity is a consequence of basic principles of special relativity

I'll put the proofs in spoilers, because some readers might want the fun of proving these things themselves, while others might be intimidated by the sight of a mathematical proof.

As a warmup, I'll start with a couple of help theorems (lemmas).


Lemma. If E₁ and E₂ are distinct events that occur at the same time in some inertial reference frame R, then E₁ and E₂ are separated by a spacelike interval.

Proof: Let the coordinates of E₁ in R be (t₁, x₁, y₁, z₁), and let the coordinates of E₂ in R be (t₂, x₂, y₂, z₂). Since E₁ and E₂ occur at the same time in R, t₁ = t₂. The coordinate-wise differences are

• dt = t₁ − t₂ = 0
• dx = x₁ − x₂
• dy = y₁ − y₂
• dz = z₁ − z₂

so the spacetime interval is

ds² = − c² dt² + dx² + dy² + dz² = dx² + dy² + dz²

That's a sum of non-negative terms. E₁ and E₂ are distinct events, so at least one of those terms must be positive, so the spacetime interval is positive.

Lemma. If Version B is true, then Version A is true.

To prove this lemma, we get to assume (temporarily!) that Version B is true.

To prove Version A follows, let E₁ and E₂ be distinct events that occur at the same time in some inertial reference frame R. By the lemma proved above, E₁ and E₂ are separated by a spacelike interval. Version B therefore tells us there is an inertial reference frame R₁ in which E₁ occurs before E₂.

The following lemma is a fact about Cartesian coordinate systems for 3-dimensional space, with the time coordinate just going along for the ride, but everyone should read the last two sentences of the spoiler.

Lemma. If E₁ and E₂ are events with coordinates (t₁, x₁, y₁, z₁) and (t₂, x₂, y₂, z₂) in some inertial reference frame R, then there is an inertial reference fram R' in which E₁ has coordinates (t₁, 0, 0, 0) and E₂ has coordinates (t₂, x₂ − x₁, 0, 0), where x₂ − x₁ > 0.

Recall that every inertial reference frame can be transformed into any other inertial reference frame via the coordinate transformations of the Poincaré symmetry. Similarly, every coordinate system that can be obtained by applying those coordinate transformations to an inertial reference frame yields another inertial reference frame.

The only coordinate transformations needed to prove this lemma are translations and rotations in space, which leave the time coordinate unchanged.

To transform R into R':

• Translate the spatial origin of R to E₁, leaving the time axis and time coordinate of E₁ unchanged.
• Rotate the spatial axes of that new coordinate system to align its positive x-axis to point along the straight line running from E₁ to E₂.

R' is the inertial reference frame that results from those two steps. E₂ lies on the x-axis of R', so its y and z coordinates are zero.

Readers who don't follow or aren't convinced by the above might benefit from this exercise in visualization.

• Grab a tape measure, some masking tape or thumbtacks, and a small rectangular box.
• Go into a rectangular room, and mark a couple of spots on some of the room's furnishings with a thumbtack or bit of masking tape. Those two points in space represent the spatial locations of E₁ and E₂.
• Use your tape measure to measure the distance from E₁ to E₂.
• Select an accessible corner of the room to represent the origin of R.
• Use your tape measure to measure how far E₁ and E₂ are from the walls that form the corner you selected and from the floor. Those six distances are the spatial coordinates of E₁ and E₂.
• Compute the differences dx, dy, and dz of those coordinates.
• Compute the sum of the squares of those differences.
• If that sum does not match the square of the distance you measured from E₁ to E₂, to within measurement error, you've done something wrong. Try again.
• Place the box snug in the corner you selected. The three edges of the box that touch the corner of the room represent the three axes of R that you're going to translate and then rotate.
• Pick up the box and, holding its sides parallel to the walls and floor as it was when you picked it up, carry it to your point E₁, so the corner of the box that was in the corner is touching E₁. You have now translated the spatial origin of R to E₁.
• Continuing to hold that corner of the box at E₁, rotate the box so the edge representing its x-axis points to E₂. You have now demonstrated to yourself how the coordinate system R gets transformed into R'.

Here is an extremely important thing to note:

That coordinate transformation didn't change the location or orientation of anything that was in the room when you started.

Here is an extremely important thing for everyone to note:

The Poincaré transformations (which include the Lorentz transformations) do not change the location or orientation or relationships between any of the events that make up the spacetime manifold. The only thing those coordinate transformations do is to change the numbers we use when specifying the coordinates of events.

Recall that the spacetime interval between two events is an invariant, which is to say that spacetime interval has the same value in all inertial reference frames. Applying that fact to the previous lemma, the spacetime interval between E₁ and E₂ has the same value in R as in R'.


Now, finally, I can prove my more general statement of the relativity of simultaneity.

Relativity of Simultaneity (Version B)

If events E₁ and E₂ are separated by a spacelike interval, then there is an inertial reference frame R₁ in which E₁ occurs before E₂, an inertial reference frame R₂ in which E₂ occurs before E₁, and an inertial reference frame R₀ in which E₁ and E₂ occur at the same time.

Suppose events E₁ and E₂ are separated by a spacelike interval in some inertial reference frame R'.

If E₁ and E₂ have the same time coordinate in R', then let R' be the R₀ we need to find.

If the time coordinate of E₁ in R' is greater than the time coordinate of E₂, then we can swap the names of those events and start over.

In what follows, therefore, we may assume the time coordinate of E₁ in R' is less than the time coordinate of E₂.

Applying the previous lemma (while swapping the names of R and R'), we can transform the coordinates of those events to an inertial reference frame R in which E₁ has coordinates (t₁, 0, 0, 0), E₂ has coordinates (t₂, x₂, 0, 0), and x₂ > 0. Because the spacetime interval is invariant,

ds² = − c² dt² + dx² > 0

where dt = t₂ − t₁ and dx = x₂, which implies

c² (t₂ − t₁)² < x₂²

which implies

c (t₂ − t₁) < x₂

(because t₁ < t₂ and x₂ > 0).

Now we can design a Lorentz transformation that will give us an inertial reference frame R₀ in which events E₁ and E₂ have the same time coordinate. Let

v = c² (t₂ − t₁) / x₂

Multiplying both sides by x₂:

v x₂ = c² (t₂ − t₁) = c (c (t₂ − t₁)) < c x₂

so v < c.

Recall that the most interesting Lorentz transformation is

t' = γ (t − vx/(c²))
x' = γ (x - vt)
y' = y
z' = z

where |v| < c and γ = 1/√(1 - (v²/(c²))). Applying that Lorentz transformation to our particular v and the coordinates of events E₁ and E₂, we get

t₁' = γ t₁
x₁' = γ (- vt₁)
y₁' = y₁ = 0
z₁' = z₁ = 0

t₂' = γ (t₂ − vx₂/(c²))
x₂' = γ (x₂ - vt₂)
y₂' = y₂ = 0
z₂' = z₂ = 0

Plugging the value of v into the equation for t₂', we get

t₂' = γ (t₂ − c² ((t₂ − t₁) / x₂) x₂ / (c²))
= γ (t₂ − (t₂ − t₁))
= γ t₁
= t₁'

so E₁ and E₂ have the same time coordinates in this new inertial reference frame, which we can take to be the R₀ we had to find.

To prove the existence of inertial reference frames R₁ and R₂ in which E₁ occurs before E₂ or E₂ occurs before E₁, we can start with R₀ and drive the time coordinates apart by applying the same kind of Lorentz transformation, once with some arbitrary positive v < c and again with some arbitrary negative v with |v| < c.

That's just some tedious high school algebra, so I'll leave it as an exercise for highly skeptical readers.

Part 3: Empirical evidence for the relativity of simultaneity

Part 2 proved the relativity of simultaneity without making any assumptions beyond basic concepts of special relativity and high school algebra. If the parts of special relativity that are used in those proofs are real, then the relativity of simultaneity must be real.

The only way the relativity of simultaneity could fail is for some of special relativity's basic ideas to be wrong.

That is extremely unlikely. Relativity is one of the most thoroughly tested theories in all of modern science. The parts of special relativity used in the above proofs have been subjected to many experimental and observational tests. Because special relativity is a special case of general relativity, all of the experimental and observational tests of general relativity also count as evidence for the correctness of special relativity, and therefore count as evidence for the relativity of simultaneity.

Although it is easy to give examples of experimental tests that the entire theory of relativity has passed, it is much harder to give specific examples of experiments that offer a direct demonstration of the relativity of simultaneity. The reason for that should be obvious. The relativity of simultaneity basically says you can't find a demonstration of simultaneity that is independent of the arbitrary reference frame(s) you choose to use in a demonstration. Furthermore, the only way to demonstrate non-simultaneity as an absolute concept (independent of reference frame) is to use events that are separated by a timelike interval, which implies all reference frames will agree on which came first.


Part 4: What do "observers" have to do with the relativity of simultaneity?

Parts 1, 2, and 3 of this essay don't mention observers at all. It was possible to state the relativity of simultaneity in mathematically precise form, and to give mathematically rigorous proofs of the relativity of simultaneity while assuming only basic principles of relativity.

That tells you the concept of "observers" is extraneous. There is no need to mention observers when discussing the relativity of simultaneity.

Yet many expositions of the relativity of simultaneity do talk about observers, especially when the exposition is aimed at a general audience.

Here is part of what a very good freshman-level textbook on special relativity has to say about observers:

Quote:

Originally Posted by A P French The literature of relativity is full of references to observers, whose role is to make judgments on the positions and times at which events occur....Almost always the observer is portrayed as being at rest with respect to one or other of two frames; by imagining an observer in each frame, one can picture an actual process for obtaining two different space-time descriptions of the same event. All this seems both harmless and reasonable....Nevertheless, the use of this language contains certain dangers.... To prevent or dispel some possible misconceptions, we add the following specific comments. 1. Although an event is by definition represented by a single point in space-time, it may nevertheless leave an enduring record of itself.... 2. ...an observer is not necessarily limited to making measurements in a reference frame to which he himself is attached....Very often, however, one will see statements such as the following: "An observer A in frame S observes that an event occurs at position x' and time t' by an observer B in frame S'." What is really being said here is just that the event has space-time coordinates (x, t) in one frame and (x', t') in the other.... 3. The last and most treacherous aspect of introducing an observer attached to a given frame of reference is that one may get the impression that this observer has some kind of bird's-eye view of the whole of his reference frame at a given instant. This is entirely false....One must be immediately on guard if one reads such colloquialisms as: "An observer attached to frame S sees the event as happening at position x and time t," or "To an observer in frame S it looks as if..." Almost always, these statements are simply statements about the space-time coordinates of a particular event as established by measurements in frame S.... The purpose of this discussion, then, is to focus the attention where it belongs—on the specification of point events according to the measures of space and time in given frames of reference....

One of the dangers not mentioned in that quotation is the danger of thinking an observer is forced to reason in terms of an inertial reference frame in which the observer is at rest. Not so. Observers are free to think in terms of any inertial frame they find convenient, and may transform back and forth between frames whenever they like. Someone traveling on a train, for example, may prefer to think of himself as being at rest while reading a book or talking to another passenger, but prefer to think of himself as moving when he looks out the window at passing scenery. The fact that you can think in terms of whatever inertial frame you like is literally the first postulate of special relativity.


Part 5: Is the relativity of simultaneity a perceptual phenomenon?

No. Emphatically no.

Parts 1, 2, and 3 of this essay were able to state and to prove the relativity of simultaneity without mentioning anything remotely connected to perception. That tells you perception has nothing at all to do with the relativity of simultaneity.

Why, then do some people make the mistake of thinking the relativity of simultaneity has something to do with perception?

Partly because some people overlook the difference between "subjective" and "perceptual". The relativity of simultaneity tells us simultaneity is either absolute and objective (in the case of events that are separated by a timelike interval) or completely relative and subjective (in the case of events separated by a spacelike interval). (In the latter case, the subjectivity comes from the fact that everyone gets to select his own preferred frame.) Perception isn't involved with either situation.

Another reason people make this mistake is the sloppy talk about observers that so many authors throw into discussions of relativity to make the exposition feel less formal and intimidating to a general audience. Even Einstein's thought experiments talked about observers, although Einstein's descriptions of those experiments were nowhere near as sloppy as some of the expositions you'll find on today's World-Wide Web.

Here is an example, written by a computer programmer who co-founded Autodesk, showing how excessive talk about observers within a somewhat confused exposition of the relativity of simultaneity can lead to a gratuitous (but fairly harmless) mention of perception:

Quote:

Originally Posted by John Walker Technical note: Some physicists prefer to reserve the term “relativity of simultaneity” for cases where observers are in motion relative to one another and the effects of special relativity obtain. In a case like the example above, where all the observers and sources are at rest with respect to one another, it is possible, using Einstein's definition of simultaneity, for spatially separated observers to synchronise their clocks and define simultaneity in terms of the spacetime interval between events. When observers are in relative motion, however, it is impossible, even in principle, to synchronise their clocks, so no definition of simultaneity is possible. But to me, the phrase “relativity of simultaneity” means precisely what it says, notwithstanding relativistic effects or their absence. In this case, three observers of the same two events see three different orders in which they appeared to occur from their particular vantage points; hence their perception of simultaneity is relative even though it is entirely due to light travel time instead of motion.

Although I'm a bit critical of Walker's exposition, my search for popular expositions of the relativity of simultaneity that described the effect as "perceptual" or a mere matter of perception turned up many that were far worse than Walker's, including several written by anti-relativity cranks.