Appendix B The Relativistic Transformation of Forces

Transcription

1 Appendix B The Relativistic Tansfomation of oces B. The ou-foce We intoduced the idea of foces in Chapte 3 whee we saw that the change in the fou-momentum pe unit time is given by the expession d d w x y. z Remembe that the fou-vecto has spatial components that ae p x c, etc. so we take the deivative with espect to w=ct to get the components of the foce. o a uantity to be a tue fou-vecto, it must tansfom accoding to the Loentz Tansfomation. Since the fou-momentum is a tue fou-vecto, must also be a fou-vecto. But when we divide this by t, we no longe have a fou-vecto, because of the way time tansfoms. If we could divide by a uantity that would be the same in all efeence fames, we would have a fou-vecto, howeve. To this end, we define pope time as time measued in the est fame of an object. (Note that pope means one s own in this context.) If we use pope time in evey fame, then all measuements of time must agee. To define the fou-foce as a tue fouvecto, all we need to do is know the elationship of pope time to the time measued in a efeence fame S. This is given in Section 5 3 as t = whee epesents the pope time and is a function of the velocity v of the paticle as measued in the fame S. Then: f d d c d d ct x y. (B ) z B. Moving Rods. Let us take two infinitely-long, coaxial, cylindical ods. The inne od is positively chaged and the oute, hollow od is negatively chaged. In a efeence fame S, the oute od is stationay and the inne od moves along the +z axis with a velocity v. A point chage is located a distance fom the od along the +x axis. It is moving with a velocity v in the +x diection. The ods have linea chage densities (chage pe unit length) of ± as measued in S. ind the foce on the chage given that the foce on fom a positive od at est is ˆx, whee is the linea chage density of the od and is a constant called the pemittivity of fee space. B

2 We know that the foce fom the negative od is just ˆx, But to find the foce fom the moving, positive od, we must fist boost the chage into the est fame of the positive od. Let us call the est fame of the positive od S and denote uantities in this fame by pimes. To tansfom to this fame, we need a boost of velocity v along the +z diection: L We can summaize the fou-vectos as follows: space-time enegy-momentum In S x pc In S x L x L p x c p c Z In this table, is a function of v and is a function of v. In S the foce on is then: Note that the linea chage density is diffeent than in S because of the contaction of the od in the S fame. Now, we wish to make a fou-foce out of this by using uation (B ) so we can tansfom the foce back into S. To do this, we fist find the time component of the fou-foce: x p x c The fou-foce is then: f x y z B

3 We now boost this back to the S fame using the invese tansfomation. L om this we can extact the thee-foce. 3 We now need to find and + in tems of uantities defined in S: And, since in S, the positive od is contacted by a facto of, a function of v, the chage must appea to be lage in S by the same facto:. Thus: 3 inally, we add this to the foce on fom the negative od to get the total foce on : 3 We see that electostatic foces of the positive and negative chages cancel out to give a esidual foce in the z diection. The oigin of a foce in the z diection is seen to be the Loentz tansfomation that mixes the t component with the z component of the fou-foce. The magnitude of this foce is smalle than the electostatic foce of a single od by a facto of, which is a vey small uantity if the velocities ae small. With a little algeba and using the identity c = / µ whee µ is anothe constant, called the pemeability of fee space, we can put this foce in an inteesting fom: 3 ẑ vv c ẑ µ dz dt v ẑv µ d µ ẑv i dt ẑv B() ẑ whee i is the cuent (chage pe unit time) caied by the positive od, and B() is the magnetic field of a long (B ) B 3

4 wie caying cuent i. In othe wods, the esidual, velocity-dependent foce esulting fom the Loentz tansfomation, is what is usually thought of as the foce esulting fom a magnetic field poduced by a moving chage. These esults can be genealized to obtain the foce on a chage moving with velocity v 3 fom a od of chage density moving with velocity v 3. The chage is a distance fom the od. ˆ c v (v ) The pat of the foce that has no velocity dependence is thought of as esulting fom an electic field,, and the pat with velocity dependence is thought of as esulting fom a magnetic field, B. In tems of fields, we have: v B ˆ B c v In fact, we can deive all of Maxwell s uations by stating fom Coulomb s Law fo souce chages at est and applying the Loentz tansfomation to foces on test chages. With this and with the assumption that electic and magnetic fields popagate at the speed of light, all of classical (not uantum) electodynamics can be geneated. Anothe way of saying the same thing is that special elativity is built into classical electodynamics. It was the appaent conflict of Maxwellian electodynamics and Newtonian mechanics that led Loentz and instein to the special theoy of elativity in the fist place. B 3. Gavitational Magnetism and Newton s Thid Law Newton s Law of Gavity and Coulomb s Law of lectostatics ae vey simila invese suae foce laws. It is clea that if electostatic foces have associated with them velocity-dependent foces because of elativistic effects, then gavitational foces must also have simila components to thei foces. We may conside these velocity dependent gavitational foces to be caused by gavitational magnetism. With electostatic foces, howeve, thee can be attaction and epulsion that cancel each othe out, leaving only the magnetic foce as a esidual foce. In the gavitational case, the effects of gavitational magnetism must be tiny compaed to the gavitational foce itself, paticulaly if bodies ae not moving nea the speed of light. Let us look at the effects of gavitational magnetism in a simple system. Thee ae two paticles in a efeence fame S. Paticle is located at the oigin and is moving in the +y diection. Paticle is located along the x axis and is moving in the +x diection. ind the foces on the paticles assuming that the coect fom fo the gavitational thee-foce on paticle when paticle is at est is: B 4

5 G ˆ whee is the est enegy of paticle, is the total enegy of paticle, and G G c 4. In the S fame, the fou-positions and fou-momenta ae: x, x x, p c, p c The Loentz tansfomations that take these to the est fame of,s, and to the est fame of, S, ae: L, L, In S, we then have: x, x x,, p x c p y c x y The foce on paticle is: G x ˆx G x ˆx o the time component of the fou-foce, we take the dot poduct of this with : G x p x c G x G x B 5

6 Combining these esults with, we can wite the fou-foce as: f x y G x z Using the invese tansfomation, we boost back to S: f G x G x inally, we can deduce the thee-foce on paticle in S: G x G The component of the foce in the y diection is the foce of gavitational magnetism. Note that it is dependent on the poduct of the velocities. Now we want to find the foce on paticle in the S fame fo compaison. We poceed in pecisely the same fashion above. In S, we have: x x, x x x, p x c p y c x y, The foce on paticle is: G x ˆx G x ˆx G x ˆx o the time component of the fou-foce, we take the dot poduct of this with : G x p x c G x B 6

7 Combining these esults with, we can wite the fou-foce as: f x y z G x G x We boost back to S: f G x G x / G x So the thee-foce on paticle in S: G x G Notice that this time thee is no foce fom gavitational magnetism. This is because paticle is located along the line of paticle 's motion. If paticle wee at some othe location along the y axis, it would expeience the effects of gavitational magnetism. Note that the x component of the foces ae also diffeent because of the diffeent s that ente each expession. Also, The foce on paticle fom paticle is not eual and opposite to the foce of paticle on paticle. Newton s Thid Law does not geneally hold in elativistic dynamics. Theefoe Newton s Thid Law does not hold fo magnetic foces, eithe. inally, we note that Newton s Law of Gavity does not hold, eithe. In fact, instein s postulate that gavitational mass and inetial mass must be euivalent led to a completely new theoy of gavitation, the Geneal Theoy of Relativity. Geneal elativity is based on the notion that matte affects the cuvatue of space and the cuvatue of space, in tun, affects the motion of matte. B 4. Some Concluding Thoughts Many people think of elativity as a set of peculia ules that goven hypothetical fast-moving objects. But we should ecognize elativity as a fundamental theoy of space, time, and motion. Its conseuences sometimes seem bizae, because they contadict ou eveyday expeience. But if we do not undestand the basics of elativity, we have a flawed pictue of natual law, even on an eveyday level. Relativity has shown us that mass end enegy can be consideed to be euivalent in some sense, and allowed us to conside the conseuences of tansfoming mass into enegy. Relativity is the cause of eveyday magnetic phenomena, and electomagnetic theoy can be undestood at a fundamental level only though elativistic pinciples. But pehaps most impotantly, special elativity gives us a tantalizing glimpse into the fundamental natue of space and time and how they ae intetwined. x B 7