4.1.1. accelerated frames

Thinking about 3-vectors from an engineering perspective, we now know that observed bookkeeper-accelerations (i.e. second time-derivatives of coordinate-position) may be caused by forces that give rise to proper-accelerations and/or by geometric-accelerations that: (i) act on every ounce of an object and (ii) vanish locally from the vantage point of a ``free float frame" which is experiencing no proper-acceleration. To see where this comes from, let's discuss briefly the origin of these concepts in the 4-vector world of general relativistic spacetime.

From the vantage point of any reference frame (accelerated or not) in any spacetime (flat and/or curved), general relativity tells us in four-vector terms (Misner1973) that momentum & energy changes per unit mass i.e. bookkeeper-accelerations (dUλ/dτ) arise directly from some combination of proper-accelerations (Aλ) that result from forces acting on a traveling object, plus from frame-dependent geometric-accelerations that act on every ounce of that object's being (-ΓλμνUμUν). From the vantage point only of unaccelerated or ``inertial" reference frames, Newton's laws further tell us that three-vector momentum changes per unit mass (dv/dt) of objects traveling at speeds small compared to ``lightspeed" need only consider the net-force (ΣF) per unit mass acting on that object, provided that we also think of gravity as another force (albeit one which acts on every ounce of our object's being).

Hence general relativity's equivalence principle (Hartle2002, Cook2004, Cheng2005) gives us the good news that Newton's second law works well locally at low speeds whether or not your frame is accelerated, as long as geometric-accelerations may be treated as proper-accelerations due to forces that act on every ounce of an object's being. In other words, questions about inertial forces in an accelerated frame, e.g. of your car when you step on the gas or take a curve at constant speed, can be treated in this context.

Example Problem 4.1.1a: You are passenger in a car accelerating at 0.5 gee from rest at a stop sign. If your bookkeeper-acceleration (in the accelerated frame of your car) is zero, then what backwards geometric-acceleration on every ounce of your being appears (in that accelerated frame) to be canceling out the forward proper-acceleration caused by the horizontal force on you from the back of your seat?

Example Problem 4.1.1b: You are in a car going at constant speed v around a leftward curve of radius R. In order to hold your bookkeeper-acceleration (in the accelerated frame of your car) to zero, what rightward geometric-acceleration on every ounce of your being must appear (in that accelerated frame) to cancel out the centripetal proper-acceleration caused by the leftward force of your car seat on you? Also in which direction would the resulting torque appear to operate, if your center of mass is located above your point of contact with the seat?

Example Problem 4.1.1c: You are pilot of an inverted aircraft going 200 mph at the top of a half-mile radius loop-the-loop. What net geometric-acceleration toward the cupholder is the coffee feeling in your now-inverted cup?

In most books, of course, only gravity's geometric-acceleration will be treated as due to a 9.8[N/kg] force-field that acts on every ounce of an object. Other local effects of geometric-acceleration are avoided by applying Newton's Laws only in inertial frames that are not experiencing motion-related geometric-acceleration.

For shell frame observers experiencing gravitational acceleration g, we might extend the local validity of Newton's 2nd law to include non-free-float frames experiencing their own proper acceleration αframe under non-relativistic conditions, where of course proper & coordinate accelerations/velocities are interchangeable.  This extended version in coordinate-notation might therefore look something like:

.

Here the proper (force-related) and geometric (kinematic) accelerations are on the left side of the equation, while the bookkeeper acceleration (e.g. of a loose piece of ice on the floor of an accelerated vehicle) is on the right. As you can see, the effect of gravity mg enters this equation in the same way whether it is conceptualized as the result of a force, or a geometric acceleration, that acts on every ounce of an object. 

Thus introducing gravity as an effect due to spacetime curvature, that may approximated as a 1/r2 force, gives students heads-up at the outset that the differential passage of time is involved, that applied science is all about strategic approximation, and that Newton's Laws may with caution even come in handy in accelerated frames. 

A second place that differential timing makes contact with everyday life (and introductory physics) is through magnetism, where once more the charge-velocities are very low but very useful effects of "the Coulomb effect" in neutral current-carrying wires still emerge. An empirical observation exercise for students is now provided here, which allows students to see how both magnetism and "low-speed" length-contraction arise simply from differences in time's passage. Since we use relativistic speeds in the simulation to illustrate this, it also allows one to get a taste of the nuanced relativistic-relationship between proper-force ΣF ≡ mα (experienced by the accelerated object) and frame-variant force Σf ≡ dp/dt.

Just as the metric equation shifts the focus from map-time t to the frame-invariant proper-time τ elapsed on the clock of any traveler, and local time-variations in turn shift the focus from coordinate-velocity v ≡ dx/dt to synchrony-free proper-velocity w ≡ dx/dτ which retains its proportionality to momentum p = mw, in studies of motion's causes the local nature of time shifts the focus from frame-variant force Σf ≡ dp/dt to the frame-invariant proper-force ΣF ≡mα representing the net external-force locally-felt by the accelerated object itself. The net proper-force is also the rate of momentum-change seen by a locally co-moving free-float-frame observer.

This net proper-force ΣF ≡ mα is not generally in the same direction as the frame-variant force Σf ≡ dp/dt, with two very robust exceptions. If the velocity-component v⊥ that's orthogonal to the proper-acceleration α direction is non-relativistic, then variant/proper force magnitudes and directions are the same and we have a relativistic version of Newton's 2nd Law for net-force, i.e. v⊥<<c implies that dp/dt ≡ Σf ≈ ΣF ≡ mα. If on the other hand only the velocity component perpendicular to the proper-acceleration direction is relativistic, then we have a magnitude-only correction as in the case of the magnetic force from a neutral current-carrying wire, i.e. v|| << c implies that dp/dt ≡ Σf ≈ ΣF/γ⊥ = mα/γ⊥. If no velocities are relativistic, of course, all of these reduce to the Newtonian case dp/dt ≡ Σf ≈ ΣF ≡ mα ≈ ma, where a ≡ dv/dt ≡ d2x/dt2 is coordinate-acceleration, since p = mdx/dτ ≈ mv. 

As far as magnitudes are concerned, in general the locally-observed net frame-variant force magnitude |Σf|  ≤ |ΣF| = mα. One can perhaps think of the source of the discrepancy (when it exists) as a proper-force dilation, which divides by γ = dt/dτ only for that component of the velocity which is transverse to the direction of motion.

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