Terminology is very important in our interpretation and presentaion of concepts. For example, we should not too loosely use terms such as mass increase when we are actually discussing secondary concepts such as change in momentum or energy, or transformations between reference frames. It is clear that the "rest mass" of an object may be determined only by one's being in the rest frame of the object. Alternatively, if we are in a frame in which that object is moving, we can measure secondary quantities, such as the energy it imaprts when colliding with a mass stationary in our reference frame. With this information, we can then calculate backwards, using whatever transforms and rules we have agreed on, and derive an inferred mass for the moving object. It is important to realize that different transforms will result in different inferred values for the mass of the object as determined from our referenc frame.

However, it is safe to assume that in the frame of the particle, its mass remains constant no matter what our velocity with respect to the particle. In special relativity, the mass of this particle increases whether it changes its velocity with respect to us (undergoes and experiences acceleration), or whether we change our velocity with respect to it. This very duality makes it clear that the mass increase is an inferred or derived property only, and is independent of the actual characteristics of the particle itself. This is where convention in terminolgy becomes important. We observe discrepancies in expected and measured energy output in collisions, and, using pre-relativistic concepts of momentum and energy, derive mass increase with velocity. The derivation is sound, even from a Newtonian standpoint, but the mass increase interpretation is suspect, as we will see.

The relation between the measured mass of an object in its rest frame and its inherent energy is well established as E₀ = m₀c² , where m₀ is referred to as the rest mass of the object. However, by convention, the total energy of an object in motion in our reference frame is referred to E = mc² , and the m in this expression is assumed to have a value different from m₀. The total energy expression contains, in addition to the rest mass of the object, some kinetic energy associated with that object’s motion through our reference frame. Since the term c is assumed constant, it would appear obvious that the value of m must have a value different than that of m₀, though it is not immediately clear whether this is just a definition of the term, or whether the mass of the object has somehow changed.

Again, following convention, we can define the momentum of the object as its energy divided by c. Thus, we have the following:

(1)

We can define p₀ as the "rest momentum" of the object, or, since it has no velocity in three space, but still moves through time, we can call this the "temporal momentum." The rationale for bringing up this convention is the following. If we consider space-time to form a four-space, then clearly (and by definition) the axis of time is normal to the three axis of space. We must be careful when adding velocities through space to motion through time to perform vector addition. Thus the magnitude of the sum of such velocities will be the square root of the sum of the squares of these velocities. Similarly, if we are adding momentums, we must use the vector sum of the orthagonal velocity vectors as well. In the frame in which the particle of (1) is moving, it has momentum due to that motion as below:

(2)

In (2), v represents the three-dimensional space vector, which, whatever its orientation in x, y and z, is normal to the time axis. Using the relations in (1), we can write an expression for the total momentum of the object in the frame in which it is moving, by vectorilly adding the momentum vectors. We then simplify to obtain a customary result:

(3)

Now, in equation (3), we have expressions for the momentum of the object in its rest frame (mc)₀, and for the momentum of the object in the frame in which it is moving (mc). Notice how the subscript on the outside of the parenthesis indicates that we are dealing with the rest value of the quantity mc, or p. The notation should not be interpreted as anything other than this. In special relativity, the convention is to assume that the subscript can be moved inside the parenthesis so that we have (mc)₀ = (m₀c), but there is no sound reason for doing so. It is not justifiable to begin with an observed value for a secondary quantity, such as momentum, and infer from the secondary quantity a definition of the components that make up the secondary quantity. In fact, in special relativity, this feat of moving the subscript inside the parenthesis is accomplished through an a priori assumption as to the validity of the Lorentz transform. This assumption then forces the transfer of the subscript inside the parenthesis, as an increase in mass is the only result compatible with the Lorentz transform. For example, we see that when David Bohm develops the expressions for mass increase with velocity, he must first assume the Lorentz transform:

Of course the assumption of the Lorentz transform is a sufficient basis for determining the formulas of special relativity. What is not stated is that the Lorentz transformations are also a necessary condition for determining the formulas of special relativity. Without the Lorentz transforms, the equations of special relativity cannot be derived. If the Lorentz transformation is incorrect, then special relativity falls as a castle built in the air. In other words, while the formulas for special relativity give correct answers for energy and momentum, it may well be that these formulas, combined with the Lorentz transformations, provide only an equivalence, not an actual picture of the true nature of the physical world which they purport to describe.

Using (3), we can express the relative energies between the rest frame of the object and the frame in which it is moving by multiplying through by c:

(4)

Let's study equation (4) for velocities such that are small compared with c. We have:

(5)

Equation (5) is simply the standard Newtonian expression for kinetic energy obtained when velocities are much less than c. When velocities become a significant fraction of c, we see that the energy imparted upon collision with an object at rest in the frame in which the test object is moving becomes substantially larger than the approximation in (5), and that simplification can no longer be applied. The results of equation (4) have been demonstrated by experemintally. Electron's traveling toward a target at a velocity approaching c are slammed into a target of aluminum. By measuring the result of the impact, one can calculate that the kinetic energy of the impact is in keeping with the results of (4). Obviously, once the electron has struck the target, and transferred all its kinetic energy to heat and material deformation, the mass of the electron in its new rest frame is the same mass it always had. Only the kinetic energy was lost in the collision.

Assume we accelerate an object of mass, m, to some arbitrarily high velocity, say 0.2c. We now allow that object to slam into a third, very massive object, traveling uniformly with respect to our reference frame at a velocity of 0.1c. In our reference frame, the object m has a certain energy, equal to the sum of its kinetic and rest energy, as measured in our reference frame. By convention, we refer to this total energy as the increased mass of the object, but it is obvious that such a convention is inappropriate.

After colliding with the large mass, the object m will be at rest in that reference frame, having dissipated all its kinetic energy in the collision. As measured by an observer in the new rest frame, the object will have a rest mass of m, even though as measured in our reference frame the object still has a total energy equal to its rest mass plus the inferred kinetic energy due to being in a reference frame that is in motion with respect to us.

Each subsequent observer in each reference frame in which the object considers itself at rest will measure the same rest mass for the object. It is clear then that the mass of the object is not changing, but only the total energy as measured with respect to any other particular reference frame.

As another demonstration, suppose that the object m never undergoes any acceleration, but instead we are accelerated away from the object to some arbitrary velocity. By convention, we again say that the mass of the object has increased, though this time it is due to our motion, and not motion imparted to the object. Any observer remaining at rest with the object will of course continue to record a mass of m for the object regardless of our velocity. We can analyze this situation from a different perspective by looking at clocks.

Thus we see that an atomic clock which has been placed in motion is susceptible to a lower frequency, and thus accumulates less time, than a clock which remains stationary for any initial rest frame of reference we choose. Note that the stationary clock does not need to be actually present. Further, the actual nature of the presumed rest frame is not important. In other words, we do not need to presume the existence of any preferred, absolutely and universally stationary frame of reference. The energy required by any cesium clock atom that has been accelerated out of its calibration reference frame and is now moving uniformly with respect to that frame will be less than that required by a clock atom which remains stationary in our reference frame, as measured in our reference frame. What we must be careful to do then, when building and testing actual atomic clocks, which must be calibrated and synchronized, is to make certain that they are all calibrated and synchronized in the same reference frame prior to starting the test. This reference frame will then become the common rest frame for all clocks in any experiment we wish to perform.

Suppose two independently moving experimenters, Alice and Bob, each construct and calibrate a clock in the respective inertial reference frames in which they reside, approaching at some fixed velocity. According to SRT, each would feel confident that their own clocks are correct, and that the other's is experiencing slowing due to motion. Assume that the clocks to be used are cesium clocks. Alice and Bob each construct and calibrate their clocks by an identical set of plans and procedures, thus the two clocks are identical except for the reference frames in which they reside. Alice uses her clock to send a signal to Bob, and Bob sends an equivalent signal to Alice. Now, since each observer knows the value of their velocity relative to the other, each can fully account for the motion induced Doppler shift, and therefore determine the effective rate of the other's clock. In this example, both Alice and Bob determine that the other's clock is ticking at exactly the same rate as their own! What happened to the slowing of clocks due to motion? The answer lies in the careful consideration of reference frames.

We have seen that clocks in motion slow down only when placed in motion relative to the rest frame in which they were constructed or calibrated. Obviously, a given cesium atom or collection of cesium atoms captured in any particular IFR will be susceptible to the same frequency, about 9 GHz.

Thus, any clock based on the cesium atom, having been constructed or calibrated in an IFR (not undergoing, or having undergone, acceleration or rotation) will keep the same time as an identical clock constructed in a different IFR. Since Alice's clock and Bob's clock each remain in their rest frame of calibration, each will record the passage of time accurately and synchronously. Recall that when a clock slows down due to being placed in motion, this has no actual effect on time itself, but only on the recording of time by that clock. The rate of the clock placed in motion becomes lower. Alice could argue that Bob's clock should be running slow, and that the reason it is not is that he has calibrated it improperly. Now we will see how the reference frames actually compare, and show why Alice cannot effectively support this argument.

We will place Alice and Bob on two identically long trains on parallel tracks, heading toward each other at very high speeds, as is illustrated in the figure. Each observer is in the front of its respective train, and carries two identical, synchronized clocks. As the trains pass each other, each observer tosses one of its clocks onto the passing train. When the last car of the passing train is along side the observer, each then tosses the clock it received back to its original train. We will assume that each of the clocks that remained with Alice and Bob in their "stationary" frame of reference recorded a time of one-hundred seconds for the trains to pass each other. After walking to the back of their respective trains and checking the clocks that were placed in motion on the passing train and then returned, each finds that these clocks recorded only ninety seconds. Now that these moving clocks are back in their initial rest frames, it is found that they are each once again marking the correct time.

Figure 1.

Alice and Bob had each already concluded that the other's clock was initially running fast. This, reasons Bob, is why Alice recorded one-hundred seconds on her clock, while his clock, traveling with her, recorded only ninety seconds. Alice reasons the same way concerning the clock she gave to Bob. This seems fine at first glance, but an apparent paradox quickly arises if we follow things a little further, exposing the fallacy of this line of reasoning.

When Bob tossed his clock onto Alice's train, it slowed down due to that acquired motion, but then sped up again upon returning to Bob's frame of reference. The clock that Alice kept with her on her train, which is also in motion with respect to Bob, kept time at a faster rate than Bob's moving clock. If Bob were to bring Alice’s clock into his frame of reference, so that in his reference frame it is no longer moving, it seems that this clock should speed up by the same amount that his own clock did when brought to rest. This would confirm his suspicion that Alice’s clocks were initially calibrated to run fast. However, we have already seen that Alice's clock, when brought into Bob's reference frame, slows down. How can this be--that one moving clock, when brought to rest, speeds up, while another slows down? The answer lies in the experimental setup--to obtain consistent results, all clocks in a given experiment must be calibrated in the same reference frame, which then becomes their common rest frame. In the above example, the clock that appeared to slow down when brought to rest was actually being placed in motion from its rest frame. Thus it was slowing due to the change in energy associated with this acquired motion--the rest frame of this clock was not the same as the clock to which it was being compared.

Alice and Bob constructed and calibrated identical clocks by identical means in two different inertial frames of reference, and found them each to be marking proper time. The reason for this is that a cesium atom is susceptible to a specific frequency. In fact, the cesium atom is used to define that frequency, rather than the other way around. As these clocks are placed in motion, they slow down due to a change in state from the reference frame in which they were calibrated. In other words, contrary to the assertions of SRT, it is not enough simply to be in motion with respect to a given reference frame. The clock must actually have been placed in motion with respect to its rest frame of calibration. It is this non-inertial change in reference frames which causes the clock to slow. As the clock is being accelerated to its final velocity, with each "tick" the clock is traveling at a new velocity with respect to its rest frame. Thus, with each "tick" the clock is running slower than it was at the previous instant. Once the clock is no longer changing its velocity with respect to its rest frame, and is moving at a constant velocity with respect to that frame, it will not continue to slow down, but will run consistently at whatever low rate it has achieved. Thus, when Alice tosses her clock into Bob's reference frame, it slows down.

Note that it is not simply an acceleration which causes a clock to slow. If such were the case, then clocks in a gravitational field or on a rotor would continually slow until they stopped, due to the constant acceleration. The only role the acceleration plays is in changing the magnitude of velocity, or speed, of the clock with respect to its rest frame. It is this relative change in energy with respect to the rest frame, caused by a change in speed, which causes the clock to run more slowly. Thus, a clock on the edge of a rotor will run at a constant slower rate than a clock at the center, despite the continual acceleration associated with circular motion. Only the acquired rotational kinetic energy has any effect on the rate of the rotor clock.

(13)