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Einstein’s special relativity and Lorentz transformation and its consequences 1. 2. 3. 4. 5. 6. Einstein’s special relativity Events and space-time in Relativity Proper time and the invariant interval Lorentz transformation Consequences of the Lorentz transformation Velocity transformation 1 1. Principle of Relativity by Einstein (1905) It is based on the following two postulates: 1) The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity), - need a transformation of coordinates which preserves the laws of physics 2) The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light. http://en.wikipedia.org/wiki/Theory_of_relativity V V A B A B Observer in the car: the light pulse reaches A and B at the same time Observer to whom the car is moving with relative V: the light pulse reaches A before B Simultaneity breaks down time cannot be regarded as a universal entity - need a different transformation from Galileo’s but will converge to it for V<<C 2 2. Events and space-time in Relativity When and where is the object under our interest An Event in Relativity. --An event is a point defined by (t, x, y, z), which describes the precise location of a “happening” which occurs at a precise point in space and at a precise time. --“Space-time” is often depicted as a “Minkowski diagram”. decelerated Time (ct) constant accelerated Space r 3 World lines The world line of an object is the unique path of that object as it travels through 4-dimensional space-time. .World lines of particles/objects at constant speed are called geodesics. Time (ct) decelerated constant accelerated Space r 4 Events and space-time in Relativity What is simultaneous in a moving frame is not simultaneous in the stationary frame. Here the signal has to be sent later ( t > 0) from A ……. 5 3 Proper time and the invariant interval 3.1 invariant interval (i.e. c is constant) In 3-dimensional EUCLIDIAN space: P1 P2 In coordinate system O: P1 ( x1 , y1 , z1 ), P2 ( x2 , y2 , z2 ) In coordinate system O’: P1 ( x'1 , y'1 , z'1 ), P2 ( x'2 , y'2 , z'2 ) r 2 x 2 y 2 z 2 x'2 y'2 z'2 In relativity, we would like to find a similar quantity for pairs of events, that is frame-independent, or the same for all observers, that is invariant interval (s)2 (ct )2 (r )2 (ct )2 (x)2 (y)2 (z)2 ∆t is the difference in time between the events ∆r is the difference between the places of occurrence of the events. 6 3.2 Events, INTERVAL AND THE METRIC A metric specifies the interval between two events 7 3.3 Proper time (length) the invariant interval The proper time between two events is the time experienced by an observer in whose frame the events take place at the same point. According to the definition of the interval between two events: c ( s ) 2 i) If (s)2 (ct )2 (r )2 0 , the interval is said to be “timelike” --there always is such a frame since positive interval means: | c | r so a frame moving at vector v = (∆r) /(∆t), in which the events take place at the same point, is moving at a speed < c ii) If (s)2 (ct )2 (r )2 0 , the interval is said to be “spacelike” --It is still invariant even though there is no frame in which both events take place at the same point. (or (c∆t)2 < 0). --There is no such frame because necessarily it would have to move faster than the speed of light. 8 Sometimes the proper distance is defined to be the distance separating two events in the frame in which they occur at the same time. It only makes sense if the interval is negative, and it is related to the interval by S 2 iii) if (s)2 (ct )2 (r )2 0 ,the interval is said to be “light-like” or null…defining a null geodesic This is the case in which i.e., (ct )2 (r )2 Or, in which the two events lie on the worldline of a photon. Because the speed of light is the same in all frames……. …. an interval equal to zero in one frame must equal zero in all frames. The three cases have different causal properties, which will be discussed later. 9 4. A transformation formula – Lorentz Transformation 4.1 The formula fits into Einstein’s two postulates We assume that relative transformation equation for x is the same as the Galileo Trans. except for a constant multiplier on the right side, i.e, x' ( x ut ) x ( x'ut ' ) (1) (2) where is a constant which can depend on u and c but not on the coordinates. (based on Postulate 1) How to find the factor ? By tracing the propagation of a light wave front in two different frames, one of which is moving with a velocity of V along x-axis w.r.t. the other. 10 Assume a light pulse that starts at the origins of S and S’ at t =t’=0 Y ’Y After a time interval the front of the wave moves Y S u Y ’ It is recorded as: S ’ (X, t) in S and (X’, t’) in S’ O ’ z Z’ O XX’ z O ’ X X’ Z’ By Einstein’s postulates 2: x = ct x’=ct’ 11 Substituting ct for x and ct’ for x’ in eqs. (1) and (2) x' ( x ut) (1) x ( x'ut' ) (2) ct ' (ct ut) (c u)t (3) ct (ct 'ut' ) (c u)t ' (4) t ' (c u ) (3) (3' ) t c t' c (4) (4' ) t (c u ) Let (3’) = (4’) u < c so is always > 1 2 If u~c, c 1 1 2 2 2 2 2 c u u 1 u / c 1 2 c 2 When u << c ~1 12 The relativistic transformation for x and and x’ is x' ( x ut) (1) x ( x'ut' ) (2) Lorentz transf. If u << c ~1 x' x ut; x x'ut Galileo transf. 13 The transformation between t and t’ can be derived: For the wave front of light, x=ct, x’=ct’ Divide c into Eq.(1) x' ct ' ( x ut) (c u)t t' c (ct ut ) (t uct ) [t 2 Divide c into Eq(2) t c (1) ux ] 2 c x ct (c u)t ' (2) ' (ct 'ut ' ) (t ' uct ' ) [t ' ux ] 2 2 c c c The complete relativistic transformation (L.T.) is ux' x ( x'ut ' ), y y' , z z ' , t [t ' 2 ] (5) c ux x' ( x ut ), y' y, z ' z, t ' [t 2 ] (6) c 14 4. 2 The interval of two events under Lorentz transformation. For two events, (t1, x1,y1,z1) and (t2,x2,y2,z2), we define: (T, X, Y, Z) = (t1-t2,x1-x2,y1-y2,z1-z2) cT ' (cT then Lorentz transformation becomes X ' (X 2 c T '2 v2 X ' [c T 2 X 2 2vXT c X 2 v 2T 2 2vXT ] 2 2 2 2 c 2T ' 2 2 (c T 2 X 2 X '2 2 v X) c v cT ) c v2 )(1 2 ), i.e., c c 2T 2 X 2 i.e., (S ' )2 (S )2 The interval of two events is an invariant under Lorentz Transformation. 15 For short: the interval is a Lorentz scalar. 4.3 Lorentz transfermation in 4-dimensional formula The L-T could be formally defined as a genernal linear transformation that leaves all intervals between any pair of events unaltered. Introduce 4-D vector ct1 x x 1 y1 z1 x ct1 , x1 , y1 , z1 S 212 (x1 x 2 )g (x1 x 2 ) 1 0 0 0 0 1 0 0 Here we have introduced: g 0 0 1 0 0 0 0 1 16 L-T can be expressed as ct ' x' y' z' X' LX v c v c 0 0 0 0 0 0 0 0 1 0 0 1 ct x y z 17