Special Relativity-General Discussion

Transcription

1 Chapter 1 Special Relativity-General Discussion Let us consider a spacetime event. By this we mean a physical occurence at some point in space at a given time. in order to characterize this event we introduce a frame of reference to measure the position and time of its occurence. A reference frame is more than just a coordinate sytem. One needs to have an apparatus (meter sticks, clocks etc) to measure r and t. Therefore by a physical frame we will mean the mathematical coordinate but with a set of apparatus attached and at rest with respect to the frame shown in Fig.1.1. This apparatus thus measures the position and time of the event. Now in relativity theory a special class of frame Inertial Frames plass a central role. These are the frames in which a particle unacted upon by force move in astraight line, i.e., they have trajectories: x(t) = r 0 + v 0 t (1.1) The significance of inertial frame was recognized by Newton in his first law which precisely mean In an inertial frame, a particle unacted upon by forces moves in a straight line and this principle was understood by Galileo. The straight line motion won t hold in all frames. 4

2 t. P(x,y,z,t) Apparatus Figure 1.1: Thus suppose one makes a coordinate transformation x = x 3, t = 1/t (1.2) x (t ) = (x 0 + v 0 /t ) 1/3 (1.3) and even though the particle is acted upon by no force it does not move in a straight line. We therefore take In an inertial frame, a particle unacted upon by forces moves in a straight line as a operational definition of an inertial frame. Now once we have found an inertial frame, can we find others? It is more or less obvious that a second frame moving with constant velocity with respect to the first will also have force free particles moving in a straight line (Fig. 1.2). Thus there are infinite number of inertial frames, which moving with relative constant velocities. This also was understood by Galileo. Now we ask the following qusetion: What is the coordiante transformation between two such inertial frames? Simple Geometry seems to include, e.g., two frames moving with relative velocity v along x-axis (Galilean transformation (Fig. 1.3): x = x vt, y = y, z = z, t = t (1.4) 5

3 t t v=const Figure 1.2: t t vt. P Figure 1.3: It is on the basis of that Newton built up his dynamics with the second law reading: d 2 r m a (t) a dt 2 = F a ( r a (t), t) (1.5) Newtonian mechanics is valid for low velocity particles. However, when particles move with velocity to c, Eq.1.4 must be changed. Einstein found that for arbtrary velocity two inertial systems are related by Lorentz transformation x = γ(x vt), y = y, z = z, t = γ(t vx/c 2 ), γ = 1 1 v 2 /c 2 (1.6) Clearly v/c 0. Eq.1.6 reduces to Eq.1.4. However Eq.1.6 contains two intersting phenomena: (i) the coefficient γ and (ii) the fact that t t. i.e., clock run differently in different 6

4 t t V Figure 1.4: inertial frames. This leads to Lorentz contractions and time dilations etc. We also want to know the form of Lorentz transformation where the forces are not simply moving parallel along the x-axis (Fig. 1.4). One can guess the answers by requiring that r is linear in r and t (or else straight line motion cannot be preserved) and also it depends on vecotor v. Thus the general form we might write is: r = f 1 (v 2 ) r + f 2 (v 2 ) vt + f 3 (v 2 )( v. r) v (1.7) t = f 4 (v 2 )t + f 5 (v 2 ) v. r, v 2 = v. v (1.8) To determine the f i, one can limit v back to being in the x-direction v = (v, 0, 0) (1.9) Then from the t equation t = f 4 t + f 5 vx (1.10) and comparing with Eq.1.6 f 4 = γ, f 5 = γ/c 2 (1.11) 7

5 For the x eqns, one has when Eq.1.9 holds x = f 1 x + f 2 vt + f 3 v 2 x, y = f 1 y, z = f 1 z (1.12) and from Eq.1.6 for y, z eqns f 1 = 1 (1.13) and for the x eqn: f 2 = γ, f 1 + f 3 v 2 = γ (1.14) Hence f 3 = (γ 1)/v 2 (1.15) transformation reads r = r γ vt + γ 1 v 2 ( v. r) v (1.16) t = γ(t v. r c 2 ) (1.17) Proper Distance and postulate of special Relativity The Lorentz transformation between two vertical frames moving with relative velocity v, e.g., Eq.1.17 are very complicated looking equations. In general, there is a much more elegant way of describing internal transformations. let us go back to pur space-time frame, and look at two points which are infinitesimally close(fig. 1.5). We define the proper distance between two points ds 2 = dx 2 + dy 2 + dz 2 c 2 (dt) 2 (1.18) 8

6 t. P (t+dt,x+dx). P(t,x) Apparatus Figure 1.5: t. P(t+dt,x+dx; t +dt, x +dx ) t. P(t,x; t, x ) Figure 1.6: In a second frame moving with relative velocity v Fig We have the same two points ds 2 = dx 2 + dy 2 + dz 2 c 2 (dt ) 2 (1.19) we can now check that if x and x are two inertial frames connected by Lorentz transformation that ds 2 = ds 2 For example the simple case of Eq.1.6 dx 2 = γ 2 (dx 2 2vdxdt + v 2 (dt) 2 ) (1.20) dt 2 = γ 2 (dt 2 2v/c 2 dxdt + v 2 /c 4 dx 2 ) (1.21) 9

7 Then dx 2 c 2 dt 2 = γ 2 (dx 2 c 2 dt 2 ) + γ 2 (v 2 dt 2 v 2 /c 2 dx 2 ) = γ 2 (1 v 2 /c 2 )(dx 2 c 2 dt 2 ) (1.22) and γ 2 (1 v 2 /c 2 ) = 1 One can also verify the more complicated case of Eq Since c always comes with dt in ds 2, it is convenient to introduce a 4th coordinate x 0 = ct (1.23) and points are specified by coordinates x µ, x µ = (x 0, x i ); µ = 0, 1, 2, 3 (1.24) and ds 2 = i (dx i ) 2 (dx 0 ) 2 (1.25) With this notation, special relativity now generalizes the rule for transformation between two inertial frames. The coordinates of a point in any two inertial frames x µ and x µ are related to each other by a linear transformation: 3 x µ = Λ µ νx ν + a µ (1.26) ν=0 Λ µ ν, a µ = const. which preserves the proper distance between points, i.e., ds 2 = ds 2 (1.27) The condition of linearlity just preserves straight line motion for a force free particles. Thus the basic condition is Eq Further this generalization says that all inertial frames are 10

8 t t a m Figure 1.7: related by Eqs.1.26 and Eq One can see that Eq.1.26 includes the Lorentz transformations we have been considering (though it is more general). Thus writing out Eq.1.26 x 0 = Λ 0 0x 0 + Λ 0 ix i + a 0 (1.28) x i = Λ i 0x 0 + Λ i jx j + a j (1.29) Comparing with Eq.1.17,. Λ 0 0 = γ, Λ 0 i = γv i /c, Λ i 0 = γv i /c, Λ i j = δ i j + γ 1 v 2 v i v j, a µ = 0 (1.30) The additional constants a µ 0 is just like the displacement between the origin of the two frames (Fig. 1.7). What is the physical significance of ds 2? Suppose two events occur simultaneously in x-frame (Fig. 1.8). Then ds 2 = dx i dx i > 0 (1.31) and for this case ds 2 = (distance) 2 between pts as measured by meter sticks w.r.t. x-frame. In the x frame we see in general from Eq.1.28 that the events do not occur at the same time 11

9 x 0. P(x i, x 0 ). P (x i +dx i,x 0 +dx 0 ) Figure 1.8: 0. P (x i +dx i,x 0 +dx 0 ) dx m. P(x i, x 0 ) Figure 1.9: x 0. P (xi,x 0 +dx 0 ) dx 0. P(x i, x 0 ) Figure 1.10: 12

10 (Fig. 1.9), i.e., dx = Λ 0 0dx 0 + Λ 0 idx i = Λ 0 idx i 0 (1.32) However, inspite of this, the apparatus using rods and clocks attached tp x frame will measure ds 2 = dx i dx i dx 0 dx 0 = ds 2 > 0 (1.33) and will get precisely the sa,e value for ds 2. Thus, for ds 2 > 0, it physically is the inavriant (distance) 2 and in the special frame where dt = 0 is actually the space distance we can say: Simultaneity is not a frame invariant concept, i.e., events that are simulataneous in one frame are not necessarily simulataneous in other inertial frames In a similar fashion, consider two events occuring at the same point in x frame (Fig1.10). Then ds 2 = (dx 0 ) 2 = c 2 dt 2 < 0 (1.34) and ( 1/c 2 ds 2 ) 1/2 is what clocks in x frame measure. However dx i = Λ i 0dx 0 + Λ i jdx j = Λ i 0dx 0 0 (1.35) and the general case is shown in Fig.1.9 where the events do not occur necessarily at the same point in the x frame. However, rods and clocks in x frame will measure ds 2 to be ds 2 = dx i dx i dx 0 dx 0 = ds 2 < 0 (1.36) Hence if ds 2 < 0, it is a generalization invariant of frame of a time interval. One commonly defines dτ 2 = 1/c 2 ds 2 = dt 2 1/c 2 dx i dx i = ( proper time interval) 2 (1.37) 13

11 and if dτ 2 > 0, dτ is the actual time interval in the frame where dx i = 0. There is a neater way of writing ds 2 introduced by Minkowski. We introduce the 4 4 matrix η (Lorentz metric) with elements: η µν = 1 +1 µ = 0 = ν µ = i = ν; i = (1.38) 0 µ ν Then clearly η is a diagonal matrix η µν = (1.39) ds 2 = dx µ η µν dx ν = dx i2 dx 0 2 (1.40) In any other inertial frame ds 2 = dx µ η µν dx ν = dx i2 dx 0 2 (1.41) ds 2 = ds 2 (1.42) and similarly dτ 2 = 1/c 2 dx µ η µν dx ν (1.43) Notation: Some authors define: η µν = (1.44) 14

12 which is negative of the previous choice. Then one would get ds 2 = dx µ η µν dx ν (1.45) dτ 2 = 1/c 2 dx µ η µν dx ν (1.46) Eqn.1.39 is called East Coast Metric and Eqn.1.44 is called West Coast Metric ] There is nothing in physics that requires the use of inertial frames, Thus suppose one makes an arbitrary coordinate transformation to an accelerated, non-inertial frame: x 0 = f 0 (x 0, x i ); x i = f i (x 0, x i ) (1.47) which we can write as x µ = f µ (x α ); or x µ = x µ (x α ) (1.48) particles unacted by forces would no longer move in a staright line. In the non-inertial frame ds 2 looks like ds 2 = dx µ η µν dx ν (1.49) dx µ = xµ x 0 dx 0 + xµ x i dx i (1.50) = xµ dx α x α dx ν = xν x β dx β (1.51) ds 2 α xµ = dx x η α µν = dx α g αβ dx β x ν x β dx β (1.52) g αβ = xµ x η x ν α µν (1.53) x β = = metric tensor 15

13 Thus the pain of using complicated non-inertial coordinates is that the simple η µν is replaced by the complicated g µν (x ). this is pure mathematics and has no physical content. One can always make the change of coordinates. However there is something that is special about inertial frames that does have physical content: The laws of physics look the same in all inertial frames ( Principle of Special Relativity ). This principle was also understood by Galileo. Th basic postulates of special relativity: (1) Definition of inertial frames (particle acted upon by forces move with constant velocity in an inertial frame) (2) Rule for coordinate transformation between inertial frames. (All inertial frames are related by a linear transformation that presreves ds 2 ) (3) Special features of inertal frames (The laws of physics look the same in all inertial frames). 16