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1.2: The Nature of Time

  • Page ID
    21803
  • Spacetime Events

    We now embark on deriving the consequences of the relativity principle in the same way that Einstein did – using a tool he called Gedankenexperiment (thought experiment). in order to keep everything straight in our discussions, we begin by defining a spacetime event.

    In the context of special relativity, a spacetime event is an instantaneous occurrence at a specific point in space and at a specific moment in time. A single point on a stationary light bulb as it dims defines a specified location, but it is not an event because the dimming process does not occur at a single instant in time. A baseball bat at exactly 12:01pm occurs at a single instant in time, but it is not an event, because the position is not specified at a single point. An easy way to visualize a spacetime event is to picture it as a very quick flash of light from a point source. The position of the point source and the instant in time the flash occurs define the spacetime “coordinates” of the spacetime event.

    It is much easier to define what a spacetime event is than it is to put physical quantities in terms of the spacetime coordinates, but as we will see, this is exactly what we will have to do to make sense of what is to come. We begin with one of most startling results, which is ironically one of the easiest to derive.

    Time Dilation

    Our first thought experiment involves turning the function of a clock into a series of spacetime events. This clock functions as follows: Light bounces back-and-forth between two mirrors, and every time it strikes one of the mirrors, the clock "ticks." We begin with Ann's perspective on what is happening with this clock. She happens to be in the same frame as the two mirror, so to her they are at rest, and the light is bouncing parallel to her \(y\)-axis. The two spacetime events we will look at are to consecutive ticks of the clock.

    Figure 1.2.1 – Ann's Perspective of the Light Clock

    ann_watches_clock.png

    Okay, so we have used the two events to determine the time span between them according to Ann. The goal of relativity is to describe what a second observer measures for a physical process given what the first observer measured, so now we introduce Bob, who is in what we called the primed inertial frame, moving at a constant speed \(v\) in the \(+x\)-direction relative to Ann. Wait this is time we are talking about! Won't both of them measure the same amount of time between ticks of the clock? Don't assume anything in relativity – just use the spacetime events and the postulate(s), and see where it leads.

    Looking from Bob's perspective means that not only is Ann moving in the \(-x\)-direction, as we noted previously, but the two events (which both occur at the top mirror) don't occur at the same position in space, since the mirror moves:

    Figure 1.2.2 – Bob's Perspective of the Light Clock

    bob_watches_clock.png

    Now we calculate the time between the two events, as we did for Ann. From Bob's perspective, the light travels a longer distance than Ann measures, and very importantly, both Ann and Bob measure the speed of light to be the same (postulate of relativity), so Bob must measure a longer time period than Ann measures between the same ticks of the light clock! According to Bob, the light travels diagonally from the top mirror to the bottom one, and the length of this half of the trip can be written in terms of the speed of light, and in terms of the pythagorean theorem:

    \[\begin{array}{l} \Delta x' = x'_2 - x'_1 = v\Delta t' \\ c\Delta t' = 2\sqrt{L^2+\left(\frac{\Delta x'}{2}\right)^2} \end{array}\]

    We can eliminate \(\Delta x'\) from these two equations to relate the time span measured by Bob to the time span measured by Ann:

    \[\Delta t' = \dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left(\dfrac{2L}{c}\right) = \gamma_v \Delta t \;,\;\;\; \gamma_v \equiv \dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}\]

    The time between ticks for Bob is greater than the time between ticks for Ann by a factor of \(\gamma_v\) (which is clearly a constant greater than 1). Just to clarify, this is not an optical illusion for Bob – he doesn't just "see the clock ticking slower than it really is," it is actually ticking slower. Also, it is important to note that while we used light to achieve this answer, it doesn't just apply to light phenomena, it applies to time flow in all its manifestations. If Ann measures her own pulse to be 60 beats per second (one second between each beat), and \(\gamma_v=2\), the Bob would measure Ann's heart rate to be 30 beats per second (2 seconds between each beat).

    It's worth taking a moment to review what the source of this result is. It comes from the fact that the light in the light clock travels farther for Bob than it does for Ann, but they agree on the speed of that light, which means that the time between the two events must be greater for Bob than it is for Ann.

    As startling as this result is, it gets weirder. Suppose Bob has a light clock exactly like Ann's. What does Ann observe when she looks at Bob's clock? She sees exactly the same thing happening with Bob's clock as he sees with her clock. Therefore Ann claims that time is passing slower for Bob than it is for her, even as Bob says that Ann's time is passing slower than his own. Which one of them is correct? Is Ann's time passing slower, or is Bob's? They are both in inertial frames, so according to the principle of relativity, each has an equal right to declare themselves to be "stationary." Therefore they are both right. The reason it seems like it is impossible that this can be true is that we cling to the incorrect notion that time is universal. The time between two events is a relative quantity that depends upon who measures it.

    Recording Spacetime Coordinates

    While the calculation above is correct, it does require an assumption that we need to briefly address. Both Ann and Bob noted the positions and times of the events in their frames. Given the importance of both position and time in relativity, we need to be specific about where these values come from. Let's consider the position coordinates first. If we just imagine an invisible three-dimensional lattice fixed in place, then when an event occurs, one just needs to note the location in the lattice.

    The time measurement is a bit trickier. If we just put a clock at the origin, then how do we know when to look at the clock. It's easy to just say "when the flash of the event occurs," but that is begging the question. We need a mechanism for noting the time on the clock. The simplest solution is to put a clock at every lattice point, so that when the flash occurs there, the clock immediately records the time. The problem is now, how do we compare times at different places? We obviously need all of our clocks to be synchronized. But again, we can't simply say they are synchronized – we need a mechanism. The way we can do it is this:

    • place a clock at the origin
    • place a clock at a lattice point
    • trigger a flash of light at the lattice point located exactly halfway between the two clocks
    • when the flash reaches a clock, set it that clock to "zero"
    • do this for the clocks at every point on the lattice

    Now we have a whole lattice of points which has its own individual timer, which is synchronized to every other clock. Now when an event occurs, we record the coordinates of the lattice point, and note the time on the clock located at that lattice point.

    Three Different Time Measurements

    Given the apparent importance of the role of measurement in determining time spans between spacetime events, we need to take a moment to distinguish between three basic types of time measurements.

    coordinate time

    Coordinate time \(\Delta t\) is the measurement of time performed by reading the times on the clocks positioned at the lattice points where the two spacetime events occur. In the example above, both Ann and Bob measured the time between ticks in their own coordinate time. For Ann, the coordinate time span between the two spacetime events was \(\Delta t = t_2 - t_1\), while for Bob it was \(\Delta t' = t'_2 - t'_1\). As we found in the thought experiment, these values are not equal, which is to say that this measurement of a time span is relative. Whenever the value of a physical quantity is different when measured from different inertial frames, we say that that quantity is frame-dependent. We therefore declare:

    Coordinate time is frame-dependent.

    proper time

    We certainly are not required to measure time between events by placing synchronized clocks at all the lattice points in our frame. Another way would be to use a single clock that is moved from the lattice point of the first event to the lattice point of the second. As before, a clock records the time of the event while it is at the same point in the lattice as the event, but this time it is the same clock, which means we do not need to rely upon our synchronization method. A time interval measured in this manner is called a proper time \(\Delta \tau\) between the spacetime events.

    Alert

    The name "proper time" dates back to the early days of relativity, and is still used today, but it is dangerously misleading for those new to the subject. The word "proper" can easily be misconstrued to mean "correct," and hopefully this section is making it clear that this is cannot be the case. We are in the process of defining three different ways of measuring the time between two events (all of which can give different answers), and none of these is any more correct than any other. The sooner the reader purges from their thoughts the notion that time is absolute and that there must be one correct value for the time between two events, the better.

    The feature that best distinguishes proper time from coordinate time is the fact that a coordinate system is not needed. Any observer in any frame can simply look at the time that flashes up on the clock's screen when it is triggered to do so by an event. This means that every observer measures the same proper time between two events. In the light clock example, one could imagine Bob watching a clock moving along with the top mirror. It is moving relative to him, so he sees it tick slower than the watch he is wearing, and in fact records the same time interval seen by Ann. We therefore conclude:

    Proper time is frame-independent.

    Besides "frame-independent," a word used to describe a physical quantity like proper time that doesn't vary from one frame to another, is invariant.

    Note that it is possible for a proper time measurement to be equal to a coordinate time measurement. For example, in the case discussed above, Ann see the two events occur at the same lattice point in her frame, so if she look at the clock place there, it is the same clock measuring the time for both events, which means it also records the proper time. Bob's measurement of coordinate time, on the other hand, is not the proper time, since he reads the numbers off two different clocks – one placed at the lattice point of the first spacetime event, and one placed at the second. From the light clock example, it should be clear that the shortest distance the light has to travel between the two mirrors occurs in Ann's frame. That is, every frame other than the "proper frame" that measures coordinate time is going to measure a longer time interval between the events than the proper time.

    spacetime interval

    This third way to measure time is actually just a very specific way of measuring proper time (so it is also an invariant). Notice that the only criterion for measuring proper time is that the same clock is present at the lattice point at the moment when the event occurs. Nowhere in this definition is it stipulated how the clock gets from on lattice point to the other. It could follow a straight line at a constant speed, as in the case of a clock positioned at the top mirror. Alternatively, it may not stay with the mirror, speeding-up and slowing-down just the right amounts along the same straight-line such that it arrives at the second event at exactly the right moment to record the second event's time. Or for that matter, the clock could be placed on a drone that starts at the top mirror when the first event occurs, then zooms all over the place, arriving back at the top mirror just in time for the second event.

    The difference between a measurement of proper time like Ann has performed and the others described above can be put very succinctly: When it is measured by a clock that follows a straight line at a constant speed, the clock remains in an inertial frame all the way from the first event to the second. Clocks that follow any other path between the two events must necessarily accelerate, and do not remain in the inertial same frame. The path that remains in an inertial frame is unique, and the proper time measured by this very specific method is called the spacetime interval \(\Delta s\). So it happens that Ann's measurement of coordinate time not only happens to be a measurement of a proper time between the two events, but it is also equal to the spacetime interval between those events.

    Let's summarize these three time measurements:

    • Measurements of coordinate and proper time follow different procedures (synchronized clocks vs. one moving clock).
    • A measurement of coordinate time happens to equal the measurement of proper time if the two events happen to occur at the same lattice point in the frame in which the coordinate time is measured.
    • A measurement of spacetime interval is a specific measurement of proper time, where the clock remains in an inertial frame during its trip from one event to the other.
    • The coordinate time, proper time, and spacetime interval are all the same value when the observer's frame is inertial (which we always assume in special relativity), and the clock remains at rest in this frame.

    Simultaneity

    Let's return to our discussion of how to measure coordinate time by synchronizing clocks at all the lattice points in a reference frame. Suppose Ann and Bob are moving past each other along the \(x\)-axis, and at the moment that their origins coincide, they start their clocks at the origin. Then each of them synchronizes all the clocks on their lattice with the clock at the origin. Doesn't this mean that all of Ann's clocks are synchronized with all of Bob's clocks? And if so, doesn't this mean that they should measure the same coordinate times between events, in contradiction to everything we have said fo far? Such a conundrum calls for a thought experiment!

    Let's suppose Ann decides to synchronize two clocks using a flash from her clock at the origin:

    Figure 1.2.3 – Simultaneous Events for Ann

    synchronizing_ann.png

    Does Bob agree that the two clocks (located at the two detectors) are synchronized? Let's look at what Bob sees:

    Figure 1.2.4 – Ann's Synchronized Events Seen by Bob (a)

    bob_views_ann_sync_1.png

    The detectors Ann is using are fixed in the lattice points in her frame, so they move along with her, according to Bob. When the light flashes, it takes time for the wave to get to the detectors, and while this time passes, the detectors move, according to Bob:

    Figure 1.2.4 – Ann's Synchronized Events Seen by Bob (b)

    bob_views_ann_sync_2.png

    As you can see, the detector trailing Ann receives the signal before the other detector, according to Bob.

    Figure 1.2.4 – Ann's Synchronized Events Seen by Bob (c)

    bob_views_ann_sync_3.png

    Far from seeing the two events simultaneously, Bob measures a time difference between them. This means that when he looks at all the clocks in Ann's lattice, he sees them all out of sync, with the times getting later the farther the clock is on the positive side of the origin. We therefore find that the concept of simultaneous events is relative (frame-dependent).

    Alert

    It is important to keep in mind that when we are talking simultaneous events in one frame, we are not talking about about simply seeing two things occur at a different time. For example, if Ann happened to be standing close to detector #1, then the light from the flag that pops up there would reach her sooner than the light coming from the flag at detector #2, and she would witness the two flags popping at different times, but the two events would still be simultaneous in her frame.

    Example \(\PageIndex{1}\)

    We found in the light clock thought experiment that the relationship between Ann's and Bob's time measurements is given by Equation 1.2.2. If Ann's two clocks are synchronized, then the time between the two events that occur when the flash reaches both detectors is zero. So why don't we find that for those same two events viewed by Bob, the time interval is also zero?

    \[\Delta t' = \gamma_v \Delta t = 0\nonumber\]

    Solution

    The equation quoted assumed that the time measured by Ann was the proper time (and the spacetime interval), since the two events occurred at the same position in her inertial frame. The synchronized events in this case do not occur at the same position, so it is the coordinate time that she measures to be zero. One way to avoid this confusion is to write the time dilation formula of Equation 1.2.2 explicitly in terms of the proper time:

    \[\Delta t' = \gamma_v \Delta \tau\nonumber\]

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