A traveler-centered intro to kinematics by P. Fraundorf of Physics & Astronomy/Center for NanoScience at University of Missouri -St Louis (63121)Treating time as a local variable permits robust approaches to
kinematics that forego questions of extended-simultaneity, which because
of their abstract nature might not be addressed explicitly until a
first relativity course and even then without considering the dependence
of clock-rates on position in a gravitational field. For example we
here use synchrony-free ``traveler kinematic" relations to construct a
brief story for beginning students about: (a) time as a local quantity
like position that depends on ``which clock", (b)
coordinate-acceleration as an approximation to the acceleration felt by a
moving traveler, and (c) the geometric origin (hence mass-independence)
of gravitational acceleration. The goal is to explicitly rule out
global-time for all from the start, so that it can be returned as a
local approximation, while tantalizing students interested in the
subject with more widely-applicable equations in range of their math
background. Sections of this paper include: Note also that figure captions may often be accessed by either mousing-over or clicking-on a given figure.## Some familiar equations re-framed:Energies associated "with the indicated frame only" include the kinetic energy of motion, as well as potential energies associated with "geometric forces" which arise in accelerated frames and in curved spacetime. Strangely enough, all of these frame-specific energies seem to connect to "differential-aging" γ ≡ dt/dτ i.e. via a relation of the form (γ-1)mc ^{2}. One thing we will try to deal with in these pages is to explore other places where (γ-1) comes in handy for describing "indicated frame-only" effects.## local time and motionAn object's momentum Greek letter "Delta" (Δ) in front of - .
where c ≈ 3×10 Thus kinetic energy is directly related to the faster passage of map (t) with respect to traveler (τ) time when simultaneity is defined by the map-frame. However, strangely enough, in non-freefloat or "geometric force" settings (where forces materialize which act on every ounce of your being) a similar relationship linked to your position rather than your speed also materializes. ## local time and rotationAn 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
Taking the derivative of this, and
dividing by m, gives a simulated g-force as a function of radius of the
form g = γ ## local time and workA 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 << c - .
Here t ## local time and gravityThe accelerated ship example above is transition from the purely kinetic centrifugal potential, and the purely motionless gravitational potential around a spherical object. where here ## local time and energy overviewGiven an energy-difference ΔE between states, one can
Thus kinetic-energy, and for geometric ("connection-coefficient") forces potential-energy and work, may all be intimately linked to ## mathematical summarykinetic minus potential energy, here written as K+U=(dt/dτ-1)mc^{2}.
Note that to the traveler in a rotating habitat, the kinetic energy
seen from the map point-of-view is a potential energy as seen from the
traveler's point-of-view. Since (dt/dτ-1)mc ^{2} (easily
obtainable from a diagonal metric) equals the map-frame's value for
kinetic minus potential energy (T-V) in the above (1+1)D examples, we've begun to explore the possibility that this quantity can also serve as a "proper-time" or traveler-point
Lagrangian L, for generating equations of motion as well as estimates
of T-V. A detailed look at L=(dt/dτ-1)mc^{2} suggests that even although the acceleration-side of the Euler-Lagrange
equation is not zero (e.g. as required for a free particle in flat spacetime) when written as a coordinate-time derivative ∂(∂L/∂w)/∂t (which works e.g. for a free particle in flat spacetime when L=T/γ), it is zero when
written as a proper-time derivative ∂(∂L/∂w)/∂τ.The non-zero force-relations that it yields in the above examples also look promising. In other words, L=T-V might qualify as a Lagrangian for describing motion in curved spacetime, even
though its generalized momentum (in the flat spacetime case p/γ = mw/γ =
mdx/dt) is not useful for tracking momentum exchanges in the map-frame,
since objects exchanging momentum generally don't share the same
proper-time. For momentum-exchanges we therefore stick with the standard
momentum p = mw = dx/dτ, which is also a proper-time derivative. The next step may be to check this out in some
classical (3+1)D cases. |