Odometers and Clocks in Introductory RelativityJohn Denker 
Abstract:The geometric approach to special relativity emphasizes the proper length (as measured by odometers) and the proper time as (measured by ordinary clocks). These properties are intrinsic to the odometers and clocks, unaffected by the motion of thisorthat reference frame.
The geometric approach treats space and time on the same footing, and treats rotations and boosts on the same footing. As a consequence we see an ordinary clock as being more similar to an odometer than to a rigid ruler.
Relativity should not be considered weird or paradoxical; it is just the geometry of spacetime. A good qualitative understanding of the essential ideas can be achieved just by using diagrams to show the correspondence between spacetime and ordinary space. This does not require much beyond highschool notions of geometry, trigonometry, and vectors.
The geometric approach stands in contrast to the socalled contraction/dilation approach, which alleges that rulers contract (relative to labframe rulers) and clocks run more slowly (relative to labframe clocks) when moving (relative to the lab frame). The geometric approach has so many advantages – in terms of simplicity, power, elegance, modernity, and consistency – that one wonders why anybody would bother with the other approach.
Additional keywords: intrinsic length, invariant length, invariant mass, chord length, chord time, inertial navigation system, twin “paradox”, FitzGeraldLorentz contraction, time dilation, time dilatation.
Note: If you are learning about relativity for the first time, please do not start here. Please read reference 1 (“Welcome to Spacetime”) before reading this document.
The following must be one of the most profound sentences in human history:
Von Stund’ an sollen Raum für sich und Zeit für sich  
völlig zu Schatten herabsinken  
und nur noch eine Art Union der beiden  
soll Selbständigkeit bewahren.  
 
From now on, space of itself and time of itself  
are to sink into mere shadows  
and only a kind of union of the two  
is to maintain its independence. 
We will make heavy use of this idea, namely the idea of spacetime. Time is the fourth dimension. Time is slightly different from the other three dimensions, but it is more the same than different. Relativity is nothing more – or less – than the geometry and trigonometry of spactime.
We shall see that this one simple idea has a huge range of profound consequences, as you can see in the righthand column of the following list:
The idea of time and the idea of threedimensional space have been around for many centuries.  Relativity gives a unified view of time and space. Time is the fourth dimension. 
The idea of a rotation in space goes back many centuries. Similarly the idea of velocity goes back many centuries.  Relativity gives a unified view of rotations in four dimensions. A velocity corresponds to a spacetime rotation (e.g. in the xt plane), and is closely analogous to an ordinary purely spatial rotation (e.g. in the xy plane). 
The idea that a velocity implies a variation of x as a function of t had been understood to first order for centuries.  The variation of t as a function of x – as required by relativity – was new and quite shocking when relativity was first understood in the early 1900s. 
The physics of energy was moreorless completely understood by 1850, and partial understanding goes back even farther than that. The idea of momentum goes back even farther. It was understood with reasonable precision by Galileo and Newton in the 1600s. 
Relativity
gives a unified view of energy and momentum.

Electrical and magnetic phenomena have been known for many centuries. The laws governing electricity and magnetism, and the relationship between the two, were known moreorless exactly by the 1860s.  Relativity gives a unified view of electromagnetism. It turns out that the Maxwell equations (1861) are fully consistent with relativity, and did not need to be corrected or updated in any way. 
Clocks, rulers, and odometers have been known for many centuries. An odometer behaves differently from a rigid ruler. 
Relativity unites the ideas of clocks and odometers.

Any one of the ideas in the righthand column would be interesting. Any two of them would be an amazing coincidence. The whole package is beyond amazing. It beggars description.
As you can see from the colorcoding, at least half of what relativity tells us is stuff we already knew. Many relativistic ideas have been known for centuries under other names, and are merely unified and systematized by relativity. Therefore you should not think that relativity is weird. Most of it is quite prosaic. You could make it seem weird if you disregarded the familiar parts and overemphasized the unfamiliar parts ... but that would be unfair and unwise, so let’s not do that.
We should also say a few words about general relativity:
Special relativity describes flat spacetime.  General relativity describes curved spacetime. 
The idea of gravity has been moreorless quantitatively understood for centuries.  The ordinary gravitational acceleration is a firstorder consequence of the local curvature of spacetime. 
If you look more closely, general relativity predicts some corrections to the Newtonian theory of gravitation. Some of these corrections have been observed. These concern such things as the gravitational interaction of spinning objects, the behavior of clocks in a gravitational potential, the notion of temperature in a gravitational potential, corrections to the orbital dynamics of planets, et cetera. 
A good introduction to spacetime can be found in reference 3. Another discussion of the correspondence between xy rotations and xt rotations can be found in reference 4.
Terminology: In this article, the term “vector” will always refer to a physical vector, which is a physical object unto itself, existing in space or spacetime. This is in contrast to any notion of a vector being defined as a list of components. If you switch from one reference frame to another, the physical vector stays the same, but the components change, as discussed in reference 5.
Terminology: The word proper as used here, in expressions such as “proper time” and “proper length”, means “its very own”, as in “proprietary”. (Other meanings of “proper”, e.g. “correct” or “appropriate”, are not relevant here.)
To understand what we mean by proper length, consider figure 1. It shows two rulers, each 12 inches long.
The ruler on the left is seen faceon by observer Joe, whose lines of sight are shown using blue, dashed lines. Joe sees the other ruler (the one on the right) somewhat endon, so that it is foreshortened by a factor of 3to1.  The ruler on the right is seen faceon by obsever Moe, whose lines of sight are shown using red, dotted lines. Mose sees the other ruler (the one on the left) somewhat endon, foreshortened by the same 3:1 factor. 
This example involves only ordinary Euclidean geometry; the analogy to relativity will become apparent soon.
The foreshortening is symmetrical, in the sense that Joe sees a foreshortened projection of Moe’s ruler, while Moe sees a foreshortened projection of Joe’s ruler.
The point here is that ordinary rulers have a wellunderstood property we call length. When we view a ruler somewhat endon, we do not conventionally say that its length has changed. We might say that the ruler appears foreshortened, or that the projection of the ruler onto our field of view is foreshortened … but “the” length is an intrinsic property of the ruler, and is invariant with respect to rotations.
We now wish to show the correspondence between ordinary geometry (typeset on the left side of the page) and relativity (typeset on the right side of the page). One might be tempted to assume that foreshortening is analogous to the FitzGeraldLorentz contraction, but we have to be careful, because that idea is “mostly correct” but not entirely correct, as we shall see.
Suppose we have a long piece of fabric, as represented by the gray area on the left side of figure 2. It is 1 meter wide, as represented by the black arrow.  Suppose we have a stick that has existed for a long time, and will continue to exist for a long time, as represented by the gray area on the right side of figure 2. The stick is 1 meter long, as represented by the black arrow. 
We lay the fabric across the table. We give it some angle θ (the Greek letter theta) in the xy plane. In the diagram, θ is on the order of 0.5 radians.  We give the stick some velocity, i.e. some angle in the xt plane, so that it is moving relative to Joe’s reference frame with some rapidity ρ (the Greek letter rho). In the diagram, ρ is on the order of 0.5. 
We make a cut parallel to the edge of the table, as shown by the heavy green bar at location y_{J} = 0. How long will the cut be? Well, it will be longer than 1 meter, i.e. more than the proper width of the fabric. This is the opposite of foreshortening! The diagonal cut is longer (not shorter) than the proper width.  We ask Joe to take a snapshot of the stick at time t_{J} = 0, as shown by the heavy green bar. What is the length of the image he records? Well, it will less than 1 meter, i.e. less than the proper length of the stick. This is the famous FitzGeraldLorentz contraction. 
We see that the FitzGeraldLorentz contraction is not closely analogous to simple foreshortening, but rather more analogous to a diagonal cut. Foreshortening involves an orthogonal projection. A diagonal cut and/or a FitzGeraldLorentz contraction can be considered a dejection, as discussed in section 3. 
If somebody asks about “the” width of the fabric, you should answer that the width is 1 meter. This is the proper width of the fabric. It is intrinsic to the fabric, independent of its orientation relative to thisorthat table.  If somebody asks about “the” length of the stick, you should answer that the length is 1 meter. This is the proper length of the stick. It is intrinsic to the stick, independent of its motion relative to thisorthat reference frame. 
When we rotate the fabric, the diagonalcut distance gets longer, but that does not mean the fabric got wider. The lengthening of the diagonal cut is not a property of the fabric; it is a property of the geometry of the situation.  When we boost the stick, the snapshot image size gets shorter, but that does not mean that the stick got shorter. The FitzGeraldLorentz contraction is not a property of the stick; it is a property of the spacetime geometry of the situation. 
Note: We are talking about a sophisticated sort of snapshot here. Joe makes use of a whole array of assistants, spread out along a contour of constant time in Joe’s frame. They answer questions about events in spacetime: Is there a piece of stick at your location right now? This sophisticated snapshot is immune from the PenroseTerrell aberration that would plague pictures taken with a simple box camera.
At present, it is alltoocommon for “popular’ accounts of relativity to suggest that the snapshot length, i.e. the FitzGeraldLorentzcontracted length, should be considered “the” length of the stick. The same goes for introductorylevel textbooks. However, we predict that over time, this will change, so that the proper length will come to be considered “the” length. This situation is analogous to the earlier shift in the meaning of “the” mass.
In relativity, there are at least three different notions that can be used to describe vectors:
All projections considered in this article are orthogonal projections, whether or not we explicitly call them orthogonal.
The rule for making an orthogonal projection is as follows: to project an object onto a given axis, carry each point of the object along a contour truly perpendicular to the axis, until it reaches the axis.
Applying this rule correctly is slightly tricky on a spacetime diagram, because the geometry of the diagramonpaper is not an entirely faithful representation of the true geometry of spacetime, as discussed in reference 4. Things that are truly perpendicular in spacetime may not look perpendicular on the diagram. For example, if you want to project something onto Moe’s xaxis (a red broken line in figure 2) you need to carry points along Moe’s contours of constant x (red solid lines in figure 2). On the right side of the diagram, they don’t look perpendicular on paper, but they are truly perpendicular in spacetime.
Specifically, if we project the heavy green bar onto Moe’s xaxis, the result corresponds to the black arrow. The reverse is not true; the heavy green bar is not the projection of the black arrow onto Joe’s xaxis. Instead we call the heavy green bar the dejection of the black arrow. (Logically and lexically this should probably be called the rejection, but mathematicians have already given that word a different geometrical meaning.)
An orthogonal projection in the xy plane always makes an image that is shorter than the original object (or possibly the same size).  An orthogonal projection in the tx plane always makes an image that is longer than the original object (or possibly the same size). Time dilation is an example of this. 
A diagonal cut is always longer than the proper width of the material. This is a dejection, not a projection onto the observer’s field of view.  A snapshot of a moving ruler is always shorter than the proper length of the ruler. The FitzGeraldLorentz contraction is an example of this. This is is a dejection, not a projection onto the observer’s field of view. 
We do not wish to have a philosophical debate about which notion of length is “real” or “correct”. It is better to ask which notions are useful under what conditions. Someone who has truly mastered the subject should be able to see things from more than one point of view, switching back and forth when necessary.
On the left side of figure 2, the proper width is at least as useful and as “real” as the length of a diagonal cut.  On the right side of figure 2, the proper length of a ruler is at least as useful and as “real” as the FitzGeraldLorentz contracted length. 
It would be unwise to accept things as “real” if and only if they can be directly measured. That’s because appearances can be deceiving. Oftentimes the important quantities can only be measured indirectly.
This point can be nicely illustrated using figure 3. This is a photograph of Saturn. The image is “real” in the sense that it is not a fake or an artist’s conception. The image of the rings is highly elliptical; specifically, the major axis is about six times as long as the minor axis. This is what our direct measurements are telling us.
Meanwhile we would say that in “real” space, the “real” rings are very nearly circular. The pronounced eccentricity of the image is a property of the image, not a property of the “real” rings. In particular the eccentricity is a property of the projection that occurred when the image was made, not a property of the “real” rings.
Let us consider the notorious “pole in a barn” problem that crops up in many textbook discussions of relativity. The scenario calls for a polevaulter carrying a pole, running at relativistic speed through a barn. The proper length of the pole is equal to the proper length of the barn. The textbook alleges that in the frame of the barn, “the” length of the pole is FitzgeraldLorentz contracted, so that the pole definitely fits inside the barn ... whereas in the frame of the runner, the textbook alleges that the barn is contracted, so that the pole definitely does not fit.
It is amusing to compare the poleandbarn system to the Saturn ring system (figure 3). In each system, there is a masochistic way and a sensible way to describe what is happening.
Saturn’s rings are made of innumerable tiny orbiting particles. A masochist could, presumably, write down equations describing the motion of the particles’ images in the plane of the image … but these equations would be bizarrely complex.  A masochist could analyze the pole and barn using the textbook approach, using the projected (i.e. FitzGeraldLorentz contracted) lengths. 
One would need to explain why the particles appear to accelerate and decelerate as they go around the ellipse. One would also need to explain why the eccentricity of the ellipse changes from month to month. 
The more sensible approach is to focus attention on the intrinsic (not projective) properties of the ring system. The description of the nearlycircular motion in the plane of the rings is incomparably simpler.  The more sensible approach is to focus attention on the intrinsic (not contracted) properties of the poleandbarn system. Draw a spacetime diagram. Don’t talk about the pole being inside the barn at a particular time, since contours of constant time differ from one reference frame to another. Instead, focus attention on fourdimensional events. 
Once we have a threedimensional description of the of the intrinsic properties of the rings, it is straightfoward to project that onto the twodimensional plane of the image.  Once we have a fourdimensional description of the intrinsic properties of the events, it is straightforward to project that onto the preferred axes of thisorthat reference frame. 
We mention this because – hitherto – people have not sufficiently appreciated the importance of intrinsic properties (as opposed to projective properties) when discussing the poleandbarn, the skateboardandmanhole, and a host of similar problems.
It is hard to imagine anyone masochistic enough to analyze ringparticles in the plane of the image.  People are being taught to do correspondingly masochistic things in spacetime. They are being taught that moving rulers contract, and moving clocks run slow. This makes relatively seem incomparably more weird and difficult than it really is. 
In summary, there are rather strong reasons why it is usually advantageous to focus attention on intrinsic properties such as proper time and proper length.
We now use what we know about length to help us understand time.
We start with a variation on the story of the notorious twins, Joe and Moe. They travel from event A to event D, each in his own car. Each car is equipped with an odometer and a clock. All the clocks and odometers are set to zero at the start of the trip.
At the end of the journey, Moe’s odometer shows a greater reading than Joe’s. There are two contrasting ways of explaining this result.
Explanation (b) is familiar to us, in our nonrelativistic travels through ordinary Euclidean space. Some paths are longer than others. Some instruments (notably rulers) are meant to be rigid, so that they measure the invariant distance between A and D. Meanwhile, other instruments (notably odometers) are meant to measure the pathlength. Other instruments (notably measuring tapes) can be used either way.
The distinction is portrayed in figure 4. Joe takes the direct path from A to D, so he can measure the length using a rigid ruler. Meanwhile, Moe takes the scenic route. The distance that he travels can be measured by an odometer, or equivalently by a flexible measuring tape that conforms to his path ABD.
As a useful bit of terminology, everyone (including Moe) can say that the ruler in figure 4 measures the chord length, i.e. the length of the chord drawn from point A to point D. Chord length stands in contrast to the proper length of Moe’s path ABD, aka the arc length. Chord length is equal to path length if and only if the path is straight; this effectively defines what we mean by “straight”.
The story of the traveling twins becomes even more interesting when we look at the clocks. Moe’s clock shows less elapsed time than Joe’s.
In the remainder of this section, we adopt description (b). We are not suggesting that description (a) is wrong; we just find description (b) vastly more convenient and more elegant.
In spacetime, clock readings are pathdependent for the same reason that odometer readings are pathdependent.
The geometry of spacetime (reference 4) is such that the scenic route will always rack up more mileage and less elapsed time than the direct route. [In some sense this defines what we mean by the direct route. This is related to the principle of least time (in geometrical optics) and the method of stationary phase (in wave mechanics and quantum mechanics), but details are beyond the scope of this article.]
This description has the virtue of consistency. In spacetime, we find it convenient to treat time on the same footing as space, and we find it convenient to treat boosts on the same footing as rotations. (reference 4) Just as viewing a ruler endon does not change “the” length, viewing a fastmoving clock does not change “the” time between clockticks. The projection onto our frame of reference may be foreshortened or dilated, but this is just a property of the projection, not a property of the rulers or clocks themselves.
Naive nonrelativistic experience suggests that the elapsed time between event A and event D is independent of the path. This is true in the nonrelativistic limit, but it is not true in general. We can say the same thing more graphically, using a spacetime diagram: elapsed time is pathindependent if all reference frames are moving along parallel worldlines, but not if the frames’ worldlines are angled relative to each other. If relativistic speeds are involved, elapsed time is pathdependent in the same way that odometer readings are pathdependent.
Many textbooks put forth an argument of the following form: “Moe’s clock racks up less elapsed time, therefore it must have been running more slowly.” This argument is invalid, since it is based on the false assumption that the correct elapsed time would be pathindependent.
Let’s be clear: Yes, at the end of the journey, Moe is younger than Joe (assuming they started out with equal ages). No, this does not prove that Moe’s clock ran slower. A far simpler explanation is that Moe took a less timeconsuming path.
Another line of argument starts by considering events called clock ticks. Each tick is an event in the technical sense of the word, meaning it occurs at a definite place and a definite time in spacetime. We take it as the defining property of a wellbehaved clock that it produces clockticks at a uniform rate, according to the contours of constant time in a reference frame at rest with respect to the clock. This definition certainly upholds the correspondence principle.
A set of such clockticks is indicated by the red circles in figure 2, near the left edge of the spacetime diagram. These were produced by a clock at rest in Moe’s frame.
The usual argument asks us to calculate
 (1) 
where τ is Moe’s proper time, t is the time in Joe’s reference frame, and ρ is once again the rapidity, namely the rapidity with which Moe’s frame is moving relative to Joe’s frame.
You can calculate Δt/Δτ geometrically using figure 2, as follows: As Moe travels from one red circle to the next, his clock ticks off one unit of “red” time, so Δτ = 1. Meanwhile, the projection of the vector onto Joe’s t axis is 13% longer than that. That is, Joe must wait more than one unit of “blue” time between the same two events.
You can also calculate Δt/Δτ algebraically.
Whether you do it geometrically and/or algebraically, everybody agrees on how to do this calculation, and everybody agrees on the numerical answer. However, alltoooften the result is encrusted with layers of metaphysics and questionable interpretations. In particular, introlevel textbooks commonly conclude from this that Moe’s clock is running slower … but any such conclusion is highly questionable. A far simpler interpretation is that Moe’s clock is keeping time just as it should, and that Δt/Δτ merely describes the projection from Moe’s frame onto Joe’s frame.
The ratio Δt/Δτ is a property of the projection, not a property of anybody’s clock. In particular, it is an orthogonal projection (like the rings in figure 3), not a dejection (like the heavy green bars in figure 2). As mentioned in section 3, projections have the property that a foreshortened ruler always appears shorter than its proper length, while the time between ticks of a moving clock always appears longer than the proper time.
In analogy to chord length, we can define a notion of chord time, namely the time ticked off by a clock that moves from point A to point D along an unaccelerated world line, i.e. a straight line on the spacetime diagram. We coin the name chordochronometer for an instrument that measures this quantity. Chord time is equal to proper time if and only if the path is unaccelerated. Reference 6 gives some formulas for calculating chord time.
The notion of “the” spacetime interval is widely used in the literature, but is ambiguous. We prefer to speak of the chord length and chord time, for reasons discussed in reference 4.
We emphasize that ruler distance and odometer distance are two welldefined, useful concepts … they just don’t happen to be the same concept. Each has a frameindependent definition (unlike Lorentzcontracted length, which pertains to a particular reference frame).
The relationship between ruler distance and odometer distance can be clearly seen if we consider a trip consisting of many steps. Let the ith step be described by a short vector, s_{i}. Then we have:
 (2) 
That is, the only difference is whether you sum before you take the norm, or take the norm before you sum. It would be silly to suggest that either of these is more “real” or more “physical” than the other. They both make sense; they’re just different.
The most complete and most “physical” description of the path is to specify all the short vectors, s_{i}. Each such vector is local to the path, intrinsic to the path, presumably directly measurable, and independent of the choice of reference frame. Given the entire sequence (s_{1}, s_{2}, ⋯), you have a complete description of the path. From there, you can summarize the data in various ways (ruler distance and/or odometer distance and/or whatever). Any such summary gives a lessthancomplete description.
As you travel along your path, you are permitted to keep a running estimate of the ruler distance and/or odometer distance … but you are not required to.
The ruler distance has the nice simplifying property that it depends only on the endpoints of the path. It is just the ordinary distance between the endpoints, which can be calculated without knowing any other details of the path.
It is straightforward to build an odometer for a wheeled vehicle; you don’t need to do much more than count the revolutions of a wheel. It is a more challenging – conceptually as well as technically – to keep track of path length and/or chord length along an arbitrary path through spacetime. The usual name for a device that does this is inertial navigation system (INS). An INS needs one brief peek at an external reference, so it can know its initial position and initial velocity, but thereafter it does not depend on external references.
An INS can be concisely described by saying it finds the velocity by integrating the acceleration, and then finds the position by integrating the velocity. That’s true, if correctly interpreted. Obviously the velocity of interest (call it v) is not Moe’s velocity relative to Moe’s reference frame, since by definition Moe’s 4velocity always has components [1, 0, 0, 0] in that frame. Instead, realize that the INS displays position relative to some chosen unaccelerated reference point, and v is Moe’s velocity relative to that point. There is no such thing as absolute position or absolute velocity, but absolute acceleration is well defined and readily measurable. Moe can integrate his acceleration to keep an uptodate estimate of v, and integrate v to get the position relative to the reference point.
The wideranging survey and analysis in reference 10 is valuable, but only tangentially connected to the topic of this article, since it concentrates on invariant (or noninvariant) notions of mass, rather than space and time.
The continued prevalence of the contracted/dilated approach is disappointing, and there does not appear to be any good reason for it. Simple inertia would be a goodenough argument in favor of triedandtrue techniques, but the contracted/dilated approach is so problematic that it must be considered triedandfoundwanting, not triedandtrue. The physicseducation research (PER) literature (e.g. reference 11) identifies “persistent difficulties” and finds that “Traditional instruction in relativity appears to have little effect on these ideas, which are present among students from the introductory to the graduate level in physics.”
On the other side of the same coin, there appears to be virtually no published PER comparing the geometric approach to the contracted/dilated approach. Reference 11 mentions Spacetime Physics (reference 3) only in connection with an elective relativity course offered to advanced students. Even then, “Both introductory and advanced courses emphasized length contraction,” leaving us with no information about the merits of emphasizing invariant length.
In the absence of systematic observations, there are powerful plausibility arguments in favor of the geometric approach, based on wellestablished pedagogical principles, such as simplicity, elegance, power, and consistency; see also item 10 below.
For what it’s worth: After the geometric approach became available, Einstein adopted it, and relied on it during the development of general relativity. That is a useful hint. Any such appeal to authority must be considered weak evidence, but it is harmless in this case, because it agrees with conclusions we have already reached via other, stronger arguments.
We are not constrained by the history of who did thisorthat. Instead, we should focus on the reasons for doing thisorthat, and judge the reasons on their merits.
Space  is to  time 
as odometer  is to  clock 
and as rigid ruler  is to  chordochronometer. 
In particular, the FitzGeraldLorentz contraction is the spacetime version of a dejection, in analogy to a diagonal cut across a piece of fabric. Similarly, time dilation is the spacetime version of a simple orthogonal projection, in analogy to the elliptical image of the rings of Saturn.
This section is something of a digression, but is included for the sake of completeness. Feel free to skip to section 9 if you wish.
In general, it is common to have two or more ways of accomplishing a given task. At one extreme, humorists including Heath Robinson, Storm P, and Rube Goldberg have invented machines to perform common tasks in ludicrously overcomplicated ways.
In physics, it is the rule rather than the exception to have more than one way of calculating the same thing. It may be that one approach is more accurate in one situation, while another approach is more accurate in another situation. In other cases, it may be merely a difference in convenience. Even when two descriptions make moreorless equivalent predictions about the observable physics, they may be quite different conceptually and pedagogically.
Historical note: The idea that rulers contract was proposed in reference 14 and reference 15 even before there was a complete theory of special relativity. In 1905, Einstein’s first relativity paper (reference 16) included the idea of clocks that run more slowly when moving. The geometric approach was developed (reference 2) a short time later, in 1908.
Scenario #1: Feynman (reference 17) presented a pair of contrasting qualitative descriptions of general relativity.
In this scenario, description (a) is little more than a parable, useful only for a limited range of pedagogical purposes. Within its limitations, its predictions are in qualitative agreement with description (b), but it cannot withstand scrutiny. For quantitative work, we are vastly better off describing general relativity in terms of invariant rulers in a curved space.
Note that these two descriptions should not be used at the same time. If you posit enough rulerdistortion to account for the physics, and additionally posit enough curvature to account for the physics, that would be overdoing it. Perhaps you could describe half the physics one way and half the other way, but this would be pointless and unnecessarily complicated.
Scenario #2: The history of special relativity furnishes another pair of contrasting descriptions:
The modern view can be understood by reference to the following very important equation:
 (3) 
where p is the 3momentum. Note that equation 3 has an elegant interpretation in terms of the dot product of the 4momentum with itself.
We see that mass and energy are not “the same thing”. Energy is energy (E), and mass is mass (m). Mass tells us the rest energy, as in equation 4. That is, E=m when p=0 – and not otherwise.
E_{rest} = m c^{2} (4) 
Tangential remark: You may have been told 100 times that energy can be «converted» or «transformed» into energy in accordance with the famous E=mc^{2} formula. That’s less than half true. In reality, in some sense rest energy can be converted to other, morenoticeable forms of energy, but equation 4 is less than half the story. The details are outside the scope of this article.
One problem with the idea of “relativistic mass” is that there are at least three inequivalent ways of defining it. One approach is to define it as E/c^{2} where E is the total energy, in contrast the invariant mass E_{rest}/c^{2} in accordance with equation 4. Using this approach, the “relativistic mass” is greater than the plain old mass by a factor of γ, where:
γ := 
 = 
 (5) 
Another attempt to define “relativistic mass” starts from the claim that mass “must” increase, because mass measures “resistance to acceleration”, and a fastmoving particle is observably more resistant to acceleration than its “rest mass” alone would predict. For straightline acceleration, we find that this “relativistic mass” is greater than the plain old mass by a factor of γ cubed. This is confusing, to say the least, because it conflicts with the previous paragraph, which involved only γ to the first power.
It is much simpler to just forget the idea of relativistic mass. As discussed in reference 18, it is perfectly possible to understand the relativistic decrease in the 3acceleration without mentioning mass at all. The 3acceleration can be seen to be smaller than the spacelike part of the proper acceleration for reasons having nothing to do with mass, merely to do with projections and the geometry of spacetime.
A third approach to “relativistic mass” uses circular motion, unlike the the straightline motion considered above. As discussed in reference 18, this gives yet another numerical value for the “relativistic mass”, conflicting with both of the previous attempts. It’s not very convenient to have a “relativistic mass” that depends on direction. Mass should be a scalar.
Given this threefold inconsistency and other problems, it makes sense to forget about “relativistic mass”. The modern, sensible approach is to describe the physics in terms of the invariant mass, proper length, and proper time. Then at the end of the day, the solution can be projected onto the laboratory frame if necessary.
In passing we remark that the shift in emphasis from (a) to (b) was accompanied by a shift in terminology. Nowadays experts (and many nonexperts as well) conventionally speak of “the” mass of the proton as being about 1 dalton. It would be redundant to call this the “rest” mass or the “invariant” mass; we just call it the mass. In contrast, some (reference 19) (but not all) early references spoke of nonconstant masses. In any case, we do not wish to argue about terminology; the contrasting ideas are more important than the conflicting terminology.
Scenario #3: We now turn away from analogies and discuss something directly relevant to the topic of this article, namely a pair of contrasting descriptions of length and time:
Either of these descriptions – if you get the details right – can be used to predict and/or describe the correct physics. However, you must be careful not to use both of them at the same time. You cannot compute the FitzGeraldLorentz contracted length and then project it onto another frame. Either the ruler gets shorter or it doesn’t; you can’t have it both ways.
When choosing which description to use, one should consider such factors as simplicity, elegance, power, and compatibility with other descriptions and models.
The way relativity has heretofore been taught to the general population – and to the next generation of physics teachers – makes things seem far more complicated than they really are. It is out of step with the way experts think about the subject, and has been for many decades. It is not technically wrong, just pedagogically inferior to the geometric approach.
We call attention to a pair of simple yet powerful ideas: treating time and space on the same footing, and treating boosts and rotations on the same footing. This gives us a nice way of visualizing special relativity:
The defining property of an ordinary clock is that it ticks at a steady rate, marking off uniform intervals of proper time. An ordinary clock carried on a journey behaves more like an odometer than a rigid ruler, recording proper time, not chord time. If you want to know the chord time, you need to carry something more complicated than an ordinary clock, i.e. something like an INS.
Thanks to Antti Savinainen, Joel Rauber, and the other members of the physl mailing list for many valuable contributions.
This article asserts “an accelerated system actually has a changed time rate” but offers no evidence of this, just prejudice and “proof” by bold assertion. It dismisses evidence to the contrary, giving no reason other than the fact that the evidence disagrees with prejudices. It fails to consider – or even mention – other explanations (such as the geometric approach) that agree with the evidence. It fails to cite the relevant prior literature.