Relativity and Cosmology Einstein and Beyond
Relativity and Cosmology Einstein and Beyond PHY 312
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This 6 page Class Notes was uploaded by Clement Bernier on Wednesday October 21, 2015. The Class Notes belongs to PHY 312 at Syracuse University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/225639/phy-312-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
Lecture 3 Review timelike and spacelike distances Review Events labelled by coordinates xt Different iner tial observers give different coordinates to same physical event Must find how to relate these different descriptions Galilean transformation not compatible with electricity and magnetism or MichelsonMorley expt New transformations called Lorentz transformations see later They are designed so that the measured speed of light is same for all observers Ensure the Principle of Relativity holds 7 Laws of Physics look same in all inertial frames With the rocket ship thought experiment we figured out that the relativity principle implied the existence of a new invariant quantity in relativistic physics 7 the distance in spacetime or spacetime interval A32 02At2 Ax Equivalently we can define the concept of proper time be tween two events Ar A3 It is clearly an invariant its value agreed upon by all inertial observers and has the dimensions of time but can we give it a more physical interpretation Suppose we are interested in two events corresponding to succes sive positions in spacetime of some material particle Suppose also that the particle is moving at constant velocity Then the proper time would be the time recorded in a frame of reference which moves with the particle so that An 0 Thus I can think about my motion in space and time as defin ing a set of events The proper time between any two of these events is then accurately recorded by the time elapsed on my wrist watch It thus has a very physical interpretation Notice A32 can be positive timelike zero and negative spacelike Material particles with nonzero mass always travel along worldlines which are timelike This ensures that the time elapsed in a co moving frame is real a co moving frame is one in which the object of interest does not move 7 Ax O in such a frame This in turn implies that there is necessarily a maximum velocity for material objects 7 c ie substitute Ax vAt into expression for spacetime distance and see that requirement that this is positive requires 1 lt 3 Light moves always along null worldlines of constant proper time Ar O Spacelike intervals separate events which can be made to be 5239 multaneous in some frame ie At 0 Two events which are spacelike separated can never causally influence each other This statement can bve proved by realizing that the existence of a causal connection between two events implies that some physi cal disturbance can propagate between them If this were so one could jump to a co moving frame and measure a proper time separating the events But that is impossible for spacelike sep arated events as the proper time would be formally the square root of negative number This division of spacetime into timelike spacelike and null sepa rated events is an important concept Consider some event and imagine drawing a spacetime diagram with that event as origin It is clear that a region within a light cone bounded by the lines i1 constitute the past and future of that event 7 that is this region contains all the timelike separated events which may had led to or follow from the event in question All other points in spacetime are spacelike separated and cannot have a causal connection to this central event The boundaries of the cone would be the worldlines of any light rays emitted or received at the event Time dilation Length Contraction 0 Suppose I have two inertial frames one of which is moving at speed 1 relative to the other We have seen from our rocket ship example that the time measured between two physical events is always shortest in the frame at which the two events take place at the same spatial position This phenomenon is called time dilation It simply follows from the formula for spacetime distance when I set Ax 0 It is important to realize that this is not a physical slowing of moving clocks Afterall from the point of the rocket ship the ship could be at rest and it is the other frame of reference which is moving Relativity implies that both points of view are equally valid Our confusion stems from the fact that we find it very difficult to break from the everyday notion of a single time independent of observer As soon as one leaves this behind all seeming problems disappear In relativity the correct way to view time is that it is just another observer dependent coordinate which has no absolute meaning For any two timelike separated events I can always find a FOR in which the time separation coordinate is smallest and the space separation coordinate is zero All types of clocks will slow exactly the same way i we age at a different rate our wristwatches run at a different rate etc etc It is just a strange effect of real physical time Example of time dilation Imagine a proton travelling at 3 40 between two detectors 2 metres apart At each detector a flash of light is emitted These constitute 2 events which are separated by 234c267c seconds between two flashes as seen from lab frame In proton frame the time separation will be A0272 2 02267C2 22 31m9t9r2 1 Thus the proton time 7 31 2 17640 Thus the time experienced by the moving proton is less than the time measured in the lab In general AU 2 At where y 18quot We see this all the time in experiments carried out on unstable high energy particles which travel close to the speed of light In a frame of reference in which the particles are not moving we observe them to decay in some characteristic time But those decays happen much more slowly as the particles are accelerated close to the speed of light This is interpreted as due to the slowing of particles internal clocks as they approach the speed of light Notice that from the particle s viewpoint what is happening is that the lab is rushing by at a speed of v From this view point then the distance between the two events must be smaller than 2m 7 since the magnitude of the relative velocity is the same and the time is shorter in the particle frame In fact the distance between the two events as seen from the particle frame is shorter by exactly the same dilation factor This is called length contraction and refers to the effect that moving objects appear contracted along their direction of motion It is yet an other indication that in relativity there is no absolute notion of space or time 7 different observers ascribe different position and time coordinates to physical events and only agree on the distance in spacetime Why 17 0 Some words of philosophy Why should I hold to this principle of relativity when it generates such a radical departure from my commonsense notions How can I be sure it is ok perhaps it is wrong We can of course take it as simply a postulate and just check whether the consequences are in accord with experiment They certainly are But maybe we can see with hindsight that it is likely to be a good physical principle Afterall in empty space there is nothing to tell me about a preferred coordinate system and I will have to discover laws of physics by simply erecting empirical frames of reference But there is nothing to tell me how to do this Any such FOR should be ok to discover the laws of Physics My friend could make a different choice and go off to discover hisher own laws of physics Who is right If there is a clear answer to this question then then all bets are off and there are no universal laws at all But if most of the Universe is empty which it is there should be no way to distinguish between different frames so logically they must all indeed be equivalent So if Physics is to get anywhere I have no option but to hope that the relativity principle is true 0 Thus the real statement of Einstein s Special Relativity is that All inertial observers are equivalent for all fundamental laws of physics Thus all new theories of physics are pretty much always constructed so as to be compatible with the relativity principle It is thus truly a cornerstone of modern physics and a logical completion of Newtonian relativity 0 Notice once more that in Special Relativity there is no absolute observer independent notion of space and time separately 7 only a funny fusion of the two i the thing we have been calling spacetime Hendeforth space by itself and time by itself are doomed to fade away into mere shadows and only a union of the two will preserve an independent reality Herman Minkowski 1864 1909
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