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General Relativity Briefing GENERAL RELATIVITY BRIEFING THE FOLLOWING TWO PAGES WILL BE FACING PAGES INSIDE THE BACK COVER Please goto next page Download file name: YInsideBackCover170421v4.pdf GENERAL RELATIVITY BRIEFING (Inside back cover, left and right sides) EVENT WORLDLINE An occurrence, such as your birth, that we can locate The path of a stone through spacetime. We can mark at a point in space at an instant of time. Events on a worldline a chain of intermediate events, such are the elemental nails on which physics hangs. An as ticks of the stone's wristwatch. The wristwatch event exists independent of any method we use to time is the reading on the stone's wristwatch at any locate it. event along its worldline. SPACETIME GLOBAL COORDINATE SYSTEM The arena in which events occur. Newton thought Any system that assigns unique numbers that time is universal, the same for all observers. (\coordinates") to every event in order to locate it Einstein realized that different observers typically in an extended spacetime region. General relativity measure different values of time between a pair of frees us to use (almost) any global coordinate system events, along with different values of their spatial in a given spacetime region. In this book we usually separation. Einstein combined space and time to analyze events that occur on a single spatial plane, provide a spacetime measure of separation between for which three coordinates, such as (t; x; y) or events on which all observers agree. [See INTERVAL (t; r; φ), suffice to locate an event. and METRIC.] FRAME A local coordinate system of our choice installed on SPACETIME REGION a patch. This local coordinate system is limited to A volume of space during a period of time, for that single patch. We may construct a frame at any example the vicinity of a black hole during one event except on a singularity such as that inside a Earth-year. black hole. SPECIAL RELATIVITY INERTIAL or FREE-FALL FRAME A theory of motion in a spacetime without gravity A local coordinate system in which special relativity or curvature. is valid, typically Cartesian spatial coordinates and GRAVITY synchronized clocks. A free stone moves with In Newtonian physics, a universal force arising from constant velocity as measured in coordinates of mass. For Newton, a force may be real (such as an inertial frame. In general relativity every inertial gravity) or fictitious (such as centrifugal force or frame is local; spacetime curvature precludes a global Coriolis force) and we can correctly analyze gravity inertial frame. in a single global inertial reference frame. [See ROOM INERTIAL FRAME.] In general relativity, however, A room is a physical enclosure of fixed spatial gravity is always a fictitious force which we can dimensions in which we make measurements and eliminate anywhere by changing to a local frame observations over an extended period of time that is in free fall (a different free-fall frame for each measured in that room. event). OBSERVER CURVATURE An observer is a person or machine that moves A measure of those properties of spacetime which through spacetime making measurements, each prevent us from having a global inertial frame. measurement limited to a local inertial frame. Thus Sources of curvature include mass-energy and an observer moves through a series of local inertial pressure. frames. WORLDTUBE GENERAL RELATIVITY A worldtube is a bundle of worldlines of objects A theory of curved spacetime and motion. at rest in a room and worldlines of the structural SINGULARITY components of that room. Think of a worldtube as A spacetime location at which curvature is so sheathing the worldline of an observer at work in the extreme that general relativity fails. room. INTERVAL PATCH Any one of three possible alternative directly- A spacetime region purposely limited in size and measured spacetime separations between any duration so that curvature does not noticeably affect pair of adjacent events. If a free stone can travel a given measurement or observation. A patch can directly from one event to the other, the time lapse have uniform gravity (which generates no curvature). on that stone's wristwatch is called the wristwatch We can approximate curved spacetime to any desired time or timelike interval. If only a light flash is accuracy with a set of patches, in the same way that fast enough to travel directly from one event to the the curved surface of the space shuttle was covered other, we say there is a null interval or lightlike with many small flat tiles. interval between them. If nothing is fast enough to STONE move between the two events, then the separation is A particle whose location at each instant is described a spacelike interval. In this case there exists an by an event and whose mass warps spacetime too inertial frame such that the two adjacent events are little to be measured. A stone has nonzero mass and simultaneous in that frame. The distance between moves slower than light with respect to every local the events measured along a ruler at rest in that inertial frame. frame is called the ruler distance. GLOBAL METRIC INVARIANT A function that satisfies Einstein's field equations, A physical quantity that has the same value, whether whose inputs are global coordinates and global measured or calculated, with respect to any possible coordinate differentials (such as dt, dr, dφ) between frame or global coordinate system. Examples include an adjacent pair of events and whose output is the the interval between two infinitesimally close events, interval between the events. The global metric (plus the wristwatch time along a worldline, the mass of the topology of the region) completely determines a stone, and the mass of a center of gravitational local spacetime and gravitational effects within the attraction. global region in which it applies. The global metric PRINCIPLE OF MAXIMAL AGING combines with the Principle of Maximal Aging [see A free stone follows the worldline of maximal entry] to predict the global motion of stones and wristwatch time (maximal aging) across any two massless particles in curved spacetime. adjoining local frames. We can completely tile spacetime with adjoining frames, so the stone knows how to move everywhere in a global spacetime region, FRAME METRIC or LOCAL METRIC such as that near a black hole or near Earth. A metric expressed in local coordinates on a CONSTANT OF MOTION spacetime patch and valid only on that patch. A A quantity that describes the motion of a free stone frame metric is always distinguished from a global or massless particle and whose value does not change metric by the notations: (1) increment ∆ instead of with time in the given global coordinate system. differential d, (2) an approximately equal sign ≈, (3) Constants of motion arise when all coefficients a one-word subscript label on every local coordinate in the metric are independent of one or more increment (such as ∆tshell) that identifies the local global coordinates. Map energy and map angular frame. In contrast, a global differential such as dt momentum are each a constant of motion for a free never carries a one-word subscript. stone near a black hole. METRIC ON A CURVED SURFACE VS. METRIC IN CURVED SPACETIME ON A STATIC CURVED SURFACE IN SPACE: IN A REGION OF CURVED SPACETIME: First, measure directly the DISTANCE between First, measure directly the SPACETIME nearby points using a ruler or tape measure| SEPARATION (timelike, lightlike, or spacelike or timed round trip of laser or radar pulse. interval) between nearby events. This direct This direct measurement requires no coordinates. measurement requires no coordinates. Second, Second, set up a SURFACE MAP, which is just set up a SPACETIME MAP, which is just a rule a rule that assigns unique coordinates to each that assigns unique coordinates to each event in point on the surface. Aside from requirements the spacetime region. Aside from some simple of uniqueness and smoothness, map coordinates requirements of uniqueness and smoothness, map are totally arbitrary. Third, ask for the relation coordinates are totally arbitrary. Third, ask between the directly-measured distance between for the relation between the directly-measured two given nearby points and their coordinate spacetime separation between two given nearby differences. The result is a local MAP SCALE events and their coordinate differences. The between the two points. Finally, generalize, result is a local MAP SCALE between the two seeking a GLOBAL METRIC: a function whose events. Finally generalize, seeking a GLOBAL inputs are coordinates and coordinate differences METRIC: a function that satisfies Einstein's between a pair of adjacent points anywhere on the field equations whose inputs are coordinates and surface and whose output is the squared distance coordinate differences between a pair of adjacent between them. events and whose output is the squared spacetime separation between them. The output of the metric is an INVARIANT: the The output of the metric is an INVARIANT: the directly-measured distance between two nearby directly-measured spacetime interval between two points has the same value no matter what map nearby events has the same value no matter what coordinates we choose. map coordinates we choose. Almost everywhere on our surface we can treat as Almost everywhere in our spacetime region we FLAT a sufficiently small surface patch. Choose treat as FLAT a sufficiently small spacetime local coordinates to turn this patch into a patch. Choose local coordinates to turn this patch FRAME. The resulting LOCAL METRIC tells us into a FRAME. The resulting LOCAL METRIC the invariant distance between any pair of points tells us the invariant interval between any pair in that frame. Demand that every measurement of events in that frame.
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