(frame 5', particle frame) f ' . fv X _ f s Figure 4.2 (frame S) f f: f). : 7' Figure 4.3 184 Chapter 4. 4evecwrs 4.5 Force and acceleration The goal in this section is to determine how forces and accelerations transform between frames, and thereby reproduce the results in Section 3.53. We'll restrict the discussion to objects with constant mass, which we' 11 call \"particles." The treatment here can be generalized to cases where the mass changes {for example, the object is being heated, or extra mass is being dumped on it), but we won't concern ourselves with these 4.5.1 Transformation of forces Let's rst look at the force 47vector in the instantaneous inertial frame S ' of a given particle. Eq. (4.10) becomes 1" = 7' (d: it") = (on (4.22) The rst component here is zero due to the reasoning in the paragraph following Eqi (4113) Equivalently. Eq. (4.22) follows from using Eq. (4.12) with a speed of Zero We will now write down two expressions for the 47force F in another frame S in which the particle moves with velocity v in the x direction. First, since F is a 4-vector, it transforms according to Eq. (4.1) So we have. using Eqi (4122). F0 = \"N"; + \"70 = Wf,',. FI = 7071' + \"\"10 = 7f; o=a= F3 = 173': fz'i (4.23) But second, from the denition in Eq. (4.10), we have F0 = y dE/ 111, F1 = 1/in F2 = 7f F3 = Yfz- (4.24) Combining Eqs. (4.23) and (4.24) yields dE/dt = v, fa = X'. fy = fv, f = fr (425) We therefore recover the results in Section 3.5.3; see Eq. (3.71). The longitudinal force is the same in both frames, but the transverse forces are larger by a factor of y in the particle's framei Hence, fyj'f,r decreases by a factor of y when going from the particle's frame to any other frame; see Fig. 4.2 and Fig. 4.3. And as a bonus, the F0 component in Eq. (4.25) tells us (after multiplying through by d! and using it dz = Jr and f; = f,) that 1113 = f), dx, which is the workeenergy result in Eq. {3.60). As noted in the rst remark in Section 31513, we can't switch the S and S' frames and write f: = fy/y. When talking about the forces on a particle, there is indeed a preferred reference frame, namely the frame .5" of the particle. All frames are not equivalent here When forming all of the 47vectors in Section 4.2. we explicitly used the dr. dz, d1, etc. for two events, and it was understood that these two events were located on the particle's worldline. 186 Chapter 4. 47 vectors There's nothing fancy going on here: the standard nonrelativistic proof of the centripetal acceleration a = vzlr works just ne again in the relativistic case. Eq. (4.8) or Eq. (4.9) then gives the 4-mcelemtion in S as A = ((1 0, ylvz/r. 0). (4.31) To nd the acceleration vectors in S' we can use the fact 5' and S are related by a Lorentz transformation in the x direction. This means that the A2 component of the 4arceleration is unchanged. So the 4acceleration in S' is also A' : A : (0, o. y2v2/r, 0). (4.32) In the particle's frame. a' Is the space part of A (using Eq. (4.8) or Er]. (4.9). With v = O and y : 1). Therefore. the 3-acceleration in 5' is a' =(0.y1v9/r. 0). (4.33) Note that our results for a and a' are consistent with Eq. (4.29). The y acceleration is larger by a factor of 3/2 in the part icle's frame . Alternatively, we can arrive at the two factors of 7/ in a' by using asimple time-dilation argument We have (this is just a repeat oqu. (3.68)) . d2)", lily , z 4) dr'z ' (Jr/W '7 4:2 2 2"2 )1 ay : y 7' (4.34) where we have used the fact that transverse lengths are the same in the two frames. 4.6 The form of physical laws The rst postulate of special relativity states that all inertial frames are equivalenll Therefore, if a physical law holds in one frame, then it must hold in all frames. Otherwise, it would be possible to differentiate between frames. As noted in the preceding section, the statement \"f = ma" cannot be a physical law. The two sides of this equation transform differently when going from one frame to another, so the statement cannot be true in all frames. If a statement has any chance of being true in all frames, it must involve only4rvectors. Consider a47vector equation (say, \"A = B") that is true in frame S. Then if we apply to this equation a Lorentz transformation (call it M) from S to another frame S', we have A = B = MA = MB =) A' = '. (4.35) The law is therefore also true in frame 5'. Of course, there are many 47vector equations that are simply not true in any frame (for example. F = P. or 2P = 3P). Only a small set of such equations (for example, F = MA) are true in at least one frame, and hence in all frames. Physical laws may also take the form of scalar equations, such as P - P = m2. A scalar is by denition a quantity that is frame independent (as we have shown the inner product to be). So if a scalar statement is true in one inertial frame, then it is true in all inertial frames Physical laws may also be higher-rank \"tensor" equations. such as the ones that arise in electromagnetism and general relativity We Won't discuss tensors here. but suice it to say that they may be thought of as things built up from 47veclorsl Scalars and 47vectors are special cases of tensors. 4.5. Force and acceleration 185 4.5.2 Transformation of accelerations The procedure here is similar to the above procedure for the force. Let's first look at the acceleration 4-vector in the instantaneous inertial frame S' of a given particle. Since ' = 0 in S', Eq. (4.8) or Eq. (4.9) gives A' = (0, a'). (4.26) We will now write down two expressions for the 4-acceleration A in another frame S in which the particle moves with velocity v in the x direction. First, since A is a 4-vector, it transforms according to Eq. (4.1). So we have, using Eq. (4.26), Ao = y(A; + VA;) = yva'", Al = y(A; + VA;) = ya'", A2 = A2 = dy, (frame S') A3 = A; = a2. (4.27) But second, from the expression in Eq. (4.9), we have ay Ao = y*vax, Al = y ax, A2 = y"dy, ax A3 = yaz. (4.28) & s Combining Eqs. (4.27) and (4.28) yields Figure 4.4 ax = a;ly, ax = a;ly, (frame S) ay = ally, az = aly. (4.29) av= ay 7/2 (The first two equations here are redundant.) We therefore again recover the results in Section 3.5.3; see Eqs. (3.68) and (3.69). So dy/ax increases by a factor of y' /y? = y when going from the particle's frame to any other frame; see Fig. 4.4 and Fig. 4.5. This ax = = ax is the opposite of the effect on fy/ fx.! This difference makes it clear that a law of the y 3 form f = ma wouldn't make any sense. If it were true in one frame, it wouldn't be true in another. Figure 4.5 Example (Acceleration for circular motion): A particle moves with constant speed v around the circle x + yz = r2, z = 0, in the lab frame. At the instant the particle crosses the negative y axis (see Fig. 4.6), find the 3-acceleration and 4-acceleration in both the lab frame and the instantaneous inertial frame of the particle (with axes chosen parallel to the lab's axes). Solution: Let the lab frame be S, and let the particle's instantaneous inertial frame be S' when it crosses the negative y axis. Then S and S' are related by a Lorentz transformation in the x direction. The 3-acceleration in S is simply a = (0, v= / r, 0). (4.30) ' In a nutshell, this difference is due to the fact that y changes with time. When talking about the acceleration Figure 4.6 4-vector, there are y's that we have to differentiate; see Eq. (4.7). This isn't the case with the force 4-vector, because the y is absorbed into the definition of p = ymy; see Eq. (4.10). This is what leads to the different powers of y in Eq. (4.28), in contrast with the identical powers in Eq. (4.24).1. Practice with 4-vectors. (40 points) In lecture we defined the 4-vectors dark, VA, A", Pu. and FA. Compute the following relativistic dot products among these 4-vectors, and verify that they are invariant by computing their value in at least two different reference frames. (Hint: a good choice for one of the reference frames is always the rest frame!) (a) V . V (b) A . A (Careful! The acceleration transforms between reference frames: see Morin Sec. 4.5.2.) (c) V . F (The forces also transform between frames, see Morin Sec. 4.5.1.) (d) F . dr, where do represents an infinitesimal displacement along the world-line of the particle on which the force acts