Time is local to a given clock, and simultaneity is determined by your choice of reference frame. Although Maxwell's equations on electromagnetism were "informed" to this reality in the mid 1800's, humans really didn't start to get the picture until the early 1900's. But how might one deal with this quantitatively?

Start with the (1+1)D flat-space metric equation, namely (cδτ)2 = (cδt)2 - (δs)2 where s (with magnitude s) and t are respectively the 3-vector position and time coordinates associated with your inertial "reference map-frame" of yardsticks and synchronized clocks. The quantity τ is the traveler or proper-time elapsed on the clocks of a traveling observer whose map-position s may be written as a function of map-time t . As usual c is the spacetime constant (literally the number of meters in a second) which is traditionally referred to as lightspeed because it is roughly the speed of light in a vacuum.

The term on the left in the metric equation is referred to as a frame-invariant. This is not a new concept. Pythagoras' theorem h2 = x2 + y2 allows one to estimate the length of a hypoteneuse h from many different x-y coordinates, all of which get the same answer. This is a (2+0)D metric equation! Similarly a given proper-time interval δτ e.g. on a traveling object's clock can be expressed in terms of δs and δt values for many different "bookkeeper" reference-frames, all of which will also agree on its duration.

The sign-change between time and space in the metric equation, not present in Pythagoras' theorem, is associated with a fundamental difference in the way time and space are experienced by our traveler. For example, distance is measured e.g. in meters with meter sticks, while time is measured e.g. in seconds with clocks.

A useful parameter in this context is the speed of proper time or "Lorentz" time-dilation factor γ ≡ δt/δτ ≥ 1 . From the metric equation, this can be shown to be γ = Sqrt[1+(w/c)2] = 1/Sqrt[1−(v/c)2] where proper velocity w ≡ δs/δτ = γv and coordinate-velocity v ≡ δs/δ t .

Proper velocity's magnitude (map distance traveled per unit time on traveler clocks) like momentum has no upper limit, but this means that coordinate-velocity's magnitude v ≤ c , since proper velocity and momentum can't be larger than infinite. When v ≪ c , then γ ≃ 1 and we can ignore the differences between t and τ, which is what we'll be doing in the Newtonian approximations of this course.

One feature of this proper time is that it is specific to only one traveler. Its meaning is therefore most useful and clear for describing the time of events local to our traveler. As we'll see later, when our traveler is accelerating or in curved spacetimes, the proper time of events far away from our traveler's "worldline" can often be defined in more than one way.

This metric equation is seriously powerful. As we see later, if one tweaks the "unit" coefficients of the terms on the right by only "one part per billion", we find ourselves in a gravitational field like that on earth where a fall of only a few meters can do you in.

As suggested before, having a new time-variable τ also gives us several new ways to measure rate of travel. In terms of 3-vectors, along with coordinate velocity v used in the Newtonian approximation, there is the synchrony-free quantity proper velocity w = γv which, as momentum per unit frame-invariant mass, has no upper limit. In terms of scalar measures of speed, there is the time-dilation factor γ = w/v , and (for unidirectional motion) the hypervelocity angle or rapidity η = arsinh[w/c] = arcosh[γ] = artanh[v/c] . Rapidity turns out to be quite useful when tracking constant felt acceleration in the unidirectional case, especially since coordinate acceleration a ≡ δv/δt can't be held constant at high speed because of coordinate velocity's upper limit.

In addition to frame-invariants like hypoteneuse s, proper time τ, and lightspeed c, another quantity with frame-invariant magnitude obtained from the metric equation is the proper acceleration α. This is nothing more than the 3-vector acceleration detected by the cell-phone accelerometer in a traveler's pocket i.e. the "felt" acceleration. 

For unidirectional motion in flat-spacetime (i.e. using map-coordinates in an inertial frame), proper-acceleration α = γ3a . These relations also yield a few simple integrals for "constant" proper-acceleration, namely α = Δw/Δt = c Δη/Δτ = c2 Δγ/Δx .

That 2nd equality involves rapidity, which as we'll see connects to proper acceleration from rest via η = ατ/c and at low speed becomes η ≃ at/c ≃ v/c ≪ 1 . The first and third equalities reduce to the familiar intro-physics integrals of constant coordinate-acceleration a ≃ Δv/Δt ≃ ½ Δ(v2)/Δx at low speed. Of course the proper acceleration equations also allow beginning students to explore accelerated round-trip problems between stars, much like the intro-physics equations give access to problems involving non-relativistic projectile trajectories on earth. 

In this unidirectional case, the flat spacetime equations of constant proper accelerated motion from rest in terms of traveler time τ then look like: x = c2/α (cosh[ατ/c] − 1) , t = c/α sinh[ατ/c] , w = c sinh[ατ/c] , γ = cosh[ατ/c], etc. These can be used to exactly solve acceleration problems in flat spacetime, e.g. like the cool (2+1)D problem shown in Figure 2 which requires messier multi-directional acceleration equations.

Note, however, that these equations are expressed in terms of proper time on the clocks of only a single traveler. When speeds are much less than c, they of reduce to the simpler equation set: x ≃ ½ ατ2 ≃ ½ at2 , t ≃ τ , w ≃ ατ ≃ v ≃ at , γ ≃ 1. These express behaviors in terms of map-time t, which Newton thought of as global, rather than in terms of one particular traveler's time τ .

We can also ask what the world looks like from the vantage point of a constant proper accelerated observer. Let's denote the apparent position e.g. of a dropped object as a function of traveler time τ as ρ[τ] .

As shown in Figure 3, there are multiple ways to describe the time and place of events far from our accelerated traveler's worldline. In that figure we illustrate with a "tangent free-float-frame" definition of extended simultaneity, and the "radar-time" definition which works more generally in curved spacetimes but is conditioned on our traveler's future as well as past. Near our traveler, traveler point dynamic equations are good approximations for these (and possibly all such) definitions of extended simultaneity.

For example, the map-position of our traveler from the constant proper acceleration equations above, for τ << c/α, becomes

x [ τ ] = ½ατ2 + 1/24 (α3/c2) τ4 + . . . ≃ ½ατ2.

From the perspective of our accelerated traveler, however, an object dropped at the τ = 0 "turn-around" point appears to recede in the opposite direction with

ρtfff[ τ ] = − (½ατ2 + 7/24 (α3/c2) τ4 + . . . ) ≃ −½ατ2 and

ρradar[ τ ] = − (½ατ2 + 10/24 (α3/c2) τ4 + . . . ) ≃ − ½ατ2.

In other words, both models of extended simultaneity allow one to locally model the dropped particle's motion as though it was accelerating opposite in direction to the direction of our observer's felt-acceleration, at about the same rate. The "traveler-point dynamic" domain of accuracy for these approximations is highlighted schematically along our traveler's worldline in yellow.

We therefore refer to such approximations, local to event origins, as traveler-point dynamic or "flat patch" approximations. In the low speed limit, both of these reduce to the Newtonian approximation: x [ t ] ≃ ½at2 and ρ [ t ] ≃ −½at2, which we'll be using in this course.

Conserved quantities like momentum and energy are important for tracking the causes of motion. Proper-velocity is connected to momentum by the simple vector relation p = mw = mγv , where m is our traveler's frame-invariant rest-mass. This reduces to p ≃ mv in the low speed limit, which we will use in this course. The upper limit of c on coordinate-speed v results simply from the fact that momentum (and proper velocity w) have to be finite.

The fact that momentum is conserved gives rise to an action-reaction theorem which involves the rate of momentum change, or net frame-variant vector force Σf = δp/δt . This theorem basically says that if system A gives momentum to system B at rate fAB, then system B in return loses momentum to system A at rate fBA = −fAB. This is Newton's "third law" that we use in this course. 

For unidirectional motion locally (and in globally-flat spacetime), the net frame-variant force on a traveler moving with proper-velocity w relates to the felt acceleration simply by Σf = mα . However, if forces act perpendicular to our traveler's velocity w then the direction of the net frame-variant force Σf differs from the direction of the felt acceleration α according to Σiξi = mα where a (perhaps differently oriented and larger) frame-invariant "felt" or proper force ξ is associated with each frame-variant force f via ξ ≡ f||w + γf⊥w where of course γ ≥ 1. However, when w ≪ c, these equations reduce to Σf ≃ ma which of course is the "second law" of Newton used in this course.

Speed of map-time γ, on the other hand, relates to conserved total energy by E = γmc2 , and to kinetic energy by K = (γ−1)mc2. This reduces to

K = ½mv2 + 3/8 (m/c2) v4 + . . . ≃ ½mv2 when v ≃ w ≪ c .

This latter is the approximation that we'll be using in this class. This relation also gives rise to an interesting result about rates of energy change in time and/or space, namely δE = Σξ ⋅ δs which reduces to the familiar relation δE = Σf ⋅ δs in unidirectional and/or low-speed applications.

In addition to point-action forces, we should also consider the effects of proper force and "geometric force" fields. The classic everyday example of the former are electrostatic and magnetic fields, which in spacetime both arise from the proper-force interaction between electric charges. The nice thing about these is that Maxwell's equations, with roots in the mid 1800's, are already informed to the rules of flat spacetime so that e.g. f = qE + qv × B, where of course here E is the vector electric field rather than scalar total energy.

In flat spacetime connection-coefficient effects, which may be approximated locally as "geometric forces" and their associated "fields", arise from choice of an accelerated reference frame. Such geometric forces act on every ounce of an object's being, are undetectable by on-board accelerometers, and vanish from the vantage point of a "free-float frame".

Common examples of these are inertial forces, like the force that pushes you back in your seat when a vehicle accelerates, the centrifugal force used to create artificial gravity environments, and the coriolis force which determines the direction of wind rotation around low pressure areas. As shown later, another such geometric force is gravity, although it arises from spacetime curvature rather than choice of an accelerated frame.

In accelerated frames (and curved spacetimes), differential aging factor γ is position dependent as well as velocity dependent. Moreover, this effect gives rise to a position-dependent potential energy for stationary objects equal to U = (γ−1)mc2 and hence a geometric force related to the gradient of this potential.

An easy to understand example of a geometric force well-depth arises in the case of rotational artificial gravity, provided at least that we ignore azimuthal forces which also operate as one's radius is changed. It is simpler because the differential aging arises purely from the tradeoff between kinetic energy in a single freefloat frame, and geometric "potential energy" whose (negative energy) well-depth to co-rotating travelers is

Wdepth = (Δtaxis/Δtoff-axis − 1)mc2 = (1/(1−(ωr/c)2) − 1)mc2 ≃ ½mω2r2.

Taking the derivative of this, and dividing by m, gives a simulated g-force as a function of radius of the form g = γ3ω2r which reduces to the classical value of ω2r = v2/r when r << c/ω. Note that the γ3 factor similarly relates proper-acceleration to coordinate-acceleration in (1+1)D flat spacetime. This "centrifugal acceleration" in an accelerated traveler's frame is perhaps the simplest case of a geometric-force (and associated "potential") linked to differential aging.

A similar type of differential aging takes place when an extended object of length L in flat spacetime undergoes constant proper-acceleration α. One obtains (at least for αL << c2, with details depending on how you define simultaneity) for the "well-depth" or work needed to climb up from the trailing to the leading edge, e.g. of your accelerating space-ship, something like:

WαL ≃ (Δtleading/Δttrailing− 1 )mc2 ≃ mαL .

Here ttrailing might also be thought of as time τ on the clocks of a passenger getting ready to "make the climb" from the trailing to the leading end of her accelerating home. This example is complicated by the fact that no single freefloat frame can be used to see this geometric "potential-energy" as a direct tradeoff with kinetic energy, as was possible in the rotation case above. This is also illustrated in Figure 3, where the "geometric force" seen by our accelerated traveler looks like f = -mα in the traveler point domain.

Like accelerated frames, gravitationally curved spacetimes (as we find here on earth) modify the flat space metric equation so that the differential-aging factor γ becomes position dependent. This in turn gives rise to geometric forces from a potential contained in (γ-1)mc2.

Consider for example a stationary traveler located a distance r outside the center of a spherically-symmetric gravitational mass M. Here the traveler's separation r, between the center of a mass-M spherical object with gravitational-acceleration grM, puts them into a potential-energy well whose depth (i.e. "escape energy") is in the limit when r is much greater than the event-horizon radius rs = 2GM/c2: 

Wesc ≃ (Δtfar/Δτ − 1)mc2 ≅ mgrMr = GmM/ r ,

where here gravity's acceleration is grM = GM/r2 and speed-of-far-time is Δtfar/Δτ = 1/Sqrt[1-2GM/(c2r)], Δtfar is time elapsed on "Schwarzschild" map-clocks far away from our object, and Δτ (also used above) is the "proper" time-elapsed on observer clocks nearer to the object surface. Put another way, acceleration due to gravity is grM = (Δtfar/Δτ-1)c2/r. In other words: On earth stuff falls because when standing "your head ages faster than your feet".

In introductory physics, we therefore imagine a force field of 9.8 N/kg near earth's surface which is constant locally, but part of a U = -GMm/r potential energy as distance from object center changes significantly. For "shell-frame" observers at fixed radius on earth's surface, this requires an ever-present upward proper acceleration of about 9.8 m/s2 which shows up on your cell-phone accelerometer, even though you are standing still.

Trajectories e.g. of dropped objects, similarly, exhibit an equal but opposite downward acceleration to shell frame observers when still near to the accelerated observer's world line, as shown in the figure at right. Note in particular that the traveler-point approximation Δr ≈ -½ατ2 works much better than the Newtonian approximation Δr ≈ -½at2, but that both work well if one is sufficiently close to the drop event itself.