Energy and Proper Time

The elapsed proper time along any specified path between two given events is the integral of dτ = √(g_mndx^mdxⁿ) along the path, with the usual summation convention, where g_mn denotes the coefficients of the metric tensor in terms of the 3+1 space-time coordinates x^a. (We can freely choose the coordinates from within the relevant diffeomorphism class, but the components of the metric depend on the chosen coordinates, and ultimately trace back to homogeneous and isotropic behavior in terms of local inertial coordinates.) The “proper time” signifies the characteristic advance of the wave function of any physical system moving along that world line.

The metric that we observe by operational measurements is actually a pseudo-metric (e.g., it does not satisfy the triangle inequality, and distinct points can be null separated), and the elapsed proper times along two different paths between the same two events are generally different. This often leads people to wonder what causes the difference. The metric arguably doesn’t have much explanatory content, it merely encodes the differential proper times along all possible paths at each point. It represents a description – not an explanation – of the metrical relations. For example, it doesn’t explain why in any sufficiently small region there are coordinates in which the metric is Minkowskian, nor why the presence of mass-energy in the vicinity is correlated with curvature of the metrical manifold.

The phrase “time dilation” is often used generically to refer to the temporal effects of relativity, both those associated with relative motion and those associated with gravitation. In 1+1 dimensional spacetime (with units so c=1), in a region with no appreciable gradients of gravitational potential, there exists inertial coordinates x,t such that the differential proper time is dτ = √[(dt)² – (dx)²], so we can integrate this along any path between two given events to give the elapsed proper time, and if we integrate along a different path we will generally get a different elapsed proper time between the same two events. This difference is the result of time dilation (or a difference in the time dilations) of the two paths. If we divide through the line element by dt we get the equivalent relation dτ/dt = √[1 – v²], which signifies the ratio by which an ideal clock moving at speed v in terms of a given system of inertial coordinates runs slow compared with the coordinate time. This too is called “time dilation”, and these are obviously just two ways of expressing the same effect.

The equivalence just noted is so obvious and trivial as to be hardly worth mentioning, except that there is a widespread misconception that these two ways of describing time dilation in special relativity represent two physically different phenomena, only one of which is to be called time dilation. This strange belief is due to the failure to recognize the essential distinction and relationship between passive and active transformations. One sees the claim that “time dilation” in special relativity refers only to passive transformations, i.e., evaluating a given interval in terms of two different systems of inertial coordinates (i.e., coordinate systems in terms of which the equations of Newtonian mechanics hold good in the low speed limit). These presentations completely miss the active transformations, meaning characteristic systems evaluated in two different states of motion in terms of a single system of inertial coordinates. It is the behavior of physical entities when subjected to changes in their states of motion that ultimately determines the relationship between systems of coordinates in terms of which inertia is homogeneous and isotropic that are the defining characteristics of inertial coordinate systems. The ratio dτ/dt is typically applied by students to passive transformations, whereas they apply the integral of dτ over dt to active transformations, without grasping the equivalence, and it prevents the students from grasping the actual epistemological basis and meaning of Lorentz invariance.

Time dilation due to relative motion is reciprocal in the sense that, given two identically constructed ideal clocks in a region with negligible variations in gravitational potential, each clock runs slow compared with the time coordinate of the inertial coordinate system in which the other clock is at rest. Nevertheless, if we have two clocks originally at rest in a single frame, and then one clock is moved away and back while the other remains at rest in that frame, the traveling clock shows less elapsed time. This leads the student to wonder why the results are not reciprocal, and what has caused the clocks to differ.

Since one clock was subject to acceleration and the other was not, the symmetry is broken, so there is no logical contradiction with reciprocity (which applies only to unaccelerated clocks), and yet this sometimes leads to confusion, because the student thinks this implies that acceleration is the cause of the time dilation, whereas it’s easy to show that the amount of time dilation is not simply proportional to the magnitude or duration of acceleration, it depends on the duration of the journey. Based on this, students often then jump to the other extreme, and think that time dilation is unrelated to acceleration. The correct explanation is that the difference in elapsed time is related to velocity and duration, or equivalently to the product of acceleration and distance. The longer the journey, the greater the distance at the turn-around point.

To illustrate, consider a clock moving from the origin of inertial coordinates at speed v for a time of T/2, then undergoing reverse acceleration 2v/Δt for a very brief period of time Δt during which its speed changes from v to –v, and then arriving back at the starting point at time T. For simplicity, we choose the  speed v low enough so that the total elapsed time for both clocks is very nearly equal to T, with no more than a very slight difference due to time dilation. In terms of the rest frame of the home clock, the traveling clock is moving at speed v for nearly the entire journey, so, neglecting the arbitrarily brief acceleration time, the traveling clock after its return will be behind the home clock by T{√[1−(v/c)²] – 1}, which to the lowest order is (1/2)(v/c)²T. On the other hand, from the standpoint of the traveling clock, the home clock is moving at speed v for nearly the entire time, so by the same reasoning one might expect that the home clock will be behind by (1/2)(v/c)²T when they are reunited… but this is impossible, because each clock cannot be behind the other. However, from the standpoint of the traveling clock there is a brief period of time Δt during which it is undergoing acceleration a = 2v/Δt while it is at a distance of L = (T/2)v from the home clock. If we regard the traveling clock as stationary, we must account for the effects of acceleration, which can be treated as a pseudo-gravitational field. The gravitational time dilation between the two clocks for this brief interval results in the traveling clock losing (aL/c²)Δt = (v/c)²T relative to the home clock. Thus, taking both effects into account, the traveling clock is behind by (1/2)(v/c)²T when they are reunited, in agreement with the expectation computed in terms of the home clock’s frame.

To give the exact result, and/or to allow for arbitrary speeds, we need to work with the full relativistic equations. For the traveling clock’s reference frame we specify a system of accelerating coordinates (during turn-around) with temporal foliations from the sequence of inertial coordinates in terms of which the traveling clock is instantaneously at rest. (This is not the only possible choice, but it is often a convenient choice.) As the clock accelerates, the foliation undergoes a tilting (relative to any fixed inertial coordinates) that accounts for the variation in the home clock’s time rate. More formally, we can write the metric coefficients for this accelerating coordinate system, and then simply integrate the proper time √(g_mndx^mdxⁿ) along the home clock’s path. It’s worth noting that need not be (and generally is not) isomorphic to a situation with a stationary gravitational field, since the equivalence principle applies only over sufficiently small regions of space and time. The notion of a uniform acceleration field over a great distance produced by a localized gravitating object is problematic. Also, if the traveling clock re-accelerates in the outward direction, the integrated proper time for the home twin as a function of this coordinate time would reverse for a sufficiently great distance and rate of acceleration. Obviously no gravitational field outside the event horizon of a localized object has this effect. This reversal is related to the fulcrum point for accelerated coordinate systems defined as above..

The example just discussed takes place in flat spacetime, although it makes use of a result from general relativity for gravitational time dilation between two locations at different gravitational potentials, combined with the equivalence principle. Notice that both kinds of relativistic time dilation – one due to motion in special relativity and the other due to different gravitational potentials in general relativity – both depend on the difference in energy between the states. The difference in elapsed time per unit time is always equal to the work (energy) necessary to move mass from one state to the other, divided by the total energy of the mass. This was discussed in another note on Time in a Centrifuge for differences in gravitational and pseudo-gravitational potential, but the same principle applies to the relativistic time dilation due to motion in special relativity. The work required to change the state of a unit mass from rest to the speed v (for fairly low speeds) divided by mc² is essentially (1/2)(v/c)², just as the work required to raise given mass a distance L in a uniform gravitational field of acceleration a (divided by the total energy) is aL/c². In the former case the effect is reciprocal, because it takes the same amount of work to accelerate an object from the rest frame of one clock to the rest frame of the other. In contrast, the gravitational case is not reciprocal, because there is an absolute sign for the difference in gravitational potential between two locations in terms of any fixed system of coordinates. (Differences in electromagnetic potential do not produce this time dilation effect, i.e., they don’t affect the spacetime metric, except to the extent that the energy serves as a source of gravitation.)

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