Newtonian spacetimeNeo-Newtonian spacetimeNewtonian spacetimeNeo-Newtonian spacetimeMotion in SpacetimeChristian Wüthrichhttp://philosophy.ucsd.edu/faculty/wuthrich/146 Philosophy of PhysicsClass 12, 6 November 2007Christian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeTaking stockNewton was (partially) successful in establishingexplanatory necessity of absolute acceleration (bucketexperiment)but he also needed absolute velocity (change of which isabsolute acceleration), which has no detectableconsequencesAs French mathematician Henri Cartan has shown in the1920s and 30s, it is possible to reformulate Newtonianmechanics without recourse to absolute velocitiesChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeNewtonian and neo-Newtonian spacetimeThis requires two steps:1formulate Newtonian mechanics in spacetime setting2replace resulting Newton spacetime with neo-Newtonian orGalilean spacetime (sometimes also “Newton-Cartanspacetime”)Christian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeDainton’s top-down route1replace the enduring points of Newton’s infinite andimmutable three-dim Euclidean continuum (“space”) with“succession of momentary and numerically distinctspacetime pointsChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetime2four-dim volume that can be regarded as a collection ofthree-dim volumes (“hyperplanes”)Christian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetime3every point is at a determinate spatial and temporaldistance from every other point; vertical lines representingpoints at different times that are at zero spatial distance(and thus in the “same place”); material objects that persistthrough a succession of spacetime points are representedby their worldlines, which, taken in their entirety, representthe object’s entire historyChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetime4distinguish absolute velocity and absolute acceleration:uniformly moving objects have straight worldlines wherethe degree of deviation from the vertical represents theabsolute velocity; curved worldlines represent objectswhich undergo absolute acceleration s.t. the steeper thecurve, the greater the acclerationChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeThe bottom-up approachmanifold of “events” (= dimensionless spacetime points)we can smoothly label events by four numbers(“coordinates”)add additional structure: time orientation, metric structureChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeA manifold of eventsChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeThe function of the metric fieldChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeWorldlines: galaxies in an expanding universeChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeSpread metric (and matter) over manifoldChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeNeo-Newtonian spacetimeNewton spt: ∃ distance relations bw points in differenthyperplanes of simultaneityneo-Newtonian spt: ¬∃ such distance relation bw points indifferent hyperplanes, only among points in eachhyperplane ( no “transtemporal” spatial distances)⇒ concept of “same place” cannot be applied over timestill have distinction bw straight and curved worldlines (viathe “affine structure” of the spt, i.e. “connection”determines which curves are straight and which ones arenot)totality of straight lines represents inertial or affinestructure of sptas notion of distance is undefined in neo-Newtonian spt,¬∃ absolute velocitiesChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeInertial transformationsTransformations bw inertial framesChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimein neo-Newtonian spt, every inertial frame can be transformedinto the rest frame:Christian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeSymmetriestransformations that leave structure of spt unchanged arecalled “symmetries” (of that spt)symmetries of hyperplanes:1rotation about a point2reflection about any axis3translation in any direction by any distanceinertial transformations are symmetries of neo-Newtonian sptChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimeComments on neo-Newtonian spacetimeNeo-Newtonian sptis not as crazy as it may at first appear (think of Huggett’squestion “How far in space are you from where you were inspace a minute ago?” (cited according to Dainton, 187)is compatible with presentism (equipped with well-definedsimultaneity relation)solves problem of undetectability of kinematic shift: ∃ no fact ofthe matter about absolute velocity, only facts about relativevelocity⇒ overcomes most ser ious Leibnizian objection (acc to Dainton) asthere are no real kinematic shifts leftbut it doesn’t solve the problem of static shiftsChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimePaul Teller’s charge of explanatory impotenceReminder: Newton’s arg from inertial effects maintained thatabsolute motion distinguished among accelerated systems andexplained why they experience inertial effectsPaul Teller: in neo-Newtonian spt, this explanation no longerworks bc absolute motion is no longer realthere, absolute acceleration is understood as the having of acurved worldlineconnection specifies which worldlines are straight/curved⇒ connection explains inertial effectsbut we specify connection, i.e. determine which worldlines arestraight/curvedthis choice occurs according to whether objects on theseworldlines experience inertial effects...⇒ circular explanation of inertial effectsChristian Wüthrich Class 12Newtonian spacetimeNeo-Newtonian spacetimePossible rebuttal of Teller’s pointWhat are inertial effects? internal stress bw parts of a bodyneo-Newtonian explanation of inertial effects: internal stressesarise when absolute velocity differences ariseHow do we detect absolute velocity changes?suppose we have object that’s a candidate for absoluteaccelerationselect some “starting” and “ending” points on its trajectoryselect an inertial reference frames for which object is at rest atstarting and ending pointsdetermine relative velocity of the two inertial frames⇒ absolute velocity change, i.e. change in velocity on which allinertial observers can agreethis gives average absolute