Relativistic Dynamics

Transcription

1 Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable ation to produe the dynamis of objets moving rapidly relative to us. It will be advantageous if that ation possess the maximum amount of symmetry. This will produe the largest number of onserved quantities whih in turn will simplify the analysis. In other words, in addition to having the usual symmetries of spae and time translation, it would be nie to have the ation be symmetri under Lorentz transformations. Remember that the lassial ations are not symmetri under Galilean transformations but are invariant instead, see Setion A.5.4. This will expand the set of onserved quantities available for the solution of dynamial problems. In the following setions, we will be more areful in our handling of the notation and remember that there are three spatial diretions, i. e. the position is x. Where it is unimportant for the interpretation, we will suppress the vetor designation The Ation for a Free Partile In order to disover the ation for rapidly moving partiles, we should look at simple situations. For the free partile, we know what the natural trajetory in spae time is a straight line. We want this to be the least ation. In addition, if we want the set of Lorentz transformations to be a symmetry for for this ation, we should onstrut it from form invariants of the Lorentz Transformations, see Setion??, and Setion 4.6. For timelike trajetories, 121

2 122 CHAPTER 7. RELATIVISTIC DYNAMICS the form invariant that haraterizes the trajetory is the proper time. The ation for the free partile should be dependent on the proper time. The simplest possibility is linearity. Sine ation has the dimensions of an energy times a time, we have to multiply the proper time by something with the dimensions of an energy. Fortunately, the relevant dimensionful parameters are available. One is the mass of the partile. In fat, when you think about it this is the definition of mass. Well, atually only that mass that is alled the inertial mass. We will expand on this idea in Chapter 8 and below in Setion 7.4. Also, sine in the ase of the free partile, the straight trajetory is the longest worldline between two events, see Setion??, the ation should be proportional to the negative of the proper time. In this way, the greatest proper time will orrespond to the least ation. The unique ombination that we have been led to is S( x 0, t 0, x f, t f ; trajetory) = m τ ( x0,t 0, x f,t f ;trajetory) (7.1) x f,t f = m Plae Figure Here trajetory, x 0,t 0 τ (7.2) This form is inappropriate for the interpretation of an ation sine it is not time slied. To transform from segment sliing whih is what we have in Equation 7.2 to time sliing, we use the fat that we an relate proper time intervals to oordinate time intervals as τ = ( t) 2 ( x)2. If we now fator out the ( t) 2 and realize that the veloity in spae time is the inverse slope to the trajetory, x t v, or x f,t f S( x 0, t 0, x f, t f ; trajetory) = m trajetory, x 0,t 0 x f,t f = m trajetory, x 0,t 0 x f,t f = m trajetory, x 0,t 0 ( t) 2 ( x)2 1 ( x) 2 ( t) 2 t 1 v2 t (7.3) 2

3 7.3. ENERGY AND MOMENTUM OF A SINGLE FREE PARTICLE123 With this time sliing, we identify the Lagrangian as L( v, x) = m 1 v2 We should ompare this result with the lassial lagrangian for the free partile, L Class ( v, x) = m v2 2. Remember, v v and v2 = v 2. In the limit the v2 1, L( v, x) = m ( ) = (m v2 2 2 m2 ). Thus, the relativisti lagrangian is the same as the lassial lagrangian to within an additive onstant. An added onstant in the lagrangian adds a term in the ation that is x f,t f trajetory, x 0,t 0 m t = m (t f t 0 ) whih does not depend on the trajetory and thus does not effet the path seletion proess. Therefore, the physis is the same for these two lagrangians in the low veloity limit. We an also alulate the relativisti free partile ation between two events over the natural path sine we already know that this is the straight line trajetory; that was how we deided what the ation was. The ation for the natural trajetory is. S( x 0, t 0, x f, t f ; natural) = m 1 ( x f x 0 ) 2 (t f t 0 ) 2 ( t f t 0 ) (7.4) 7.3 Energy and momentum of a single free partile Using the fat that the spatial and temporal translations are a ontinuous symmetry for this ation, we have energy and momentum onservation. Using Noether s theorem, Setion??, the energy is the hange in the ation when the final time is shifted or E = δs δt f (7.5) = 1 m sine for the straight line trajetory, x f x 0 t f t 0 = ( x f x 0 ) 2 (t f t 0 ) 2 (7.6) m. (7.7) = v.

4 124 CHAPTER 7. RELATIVISTIC DYNAMICS Similarly, the momentum is the hange in the ation when you translate the final position. p = δs δ x f (7.8) = = m ( x f x 0 ) (t f t 0 ) 1 ( x f x 0 ) 2 (t f t 0 ) 2 (7.9) m v (7.10) Note that, for a massive partile observed in its rest frame, v = 0, the momentum is zero and the energy is m. Thus this energy is not neessarily an energy of motion like the lassial kineti energy, see Setion 7.5. On the other hand, note that, for a massive partile that is moving relative to some frame at a relative speed v, the energy and momentum are dependent on the relative motion of the observer. This is how it was in old fashioned lassial physis; the momentum was p = m v and the energy of motion or kineti energy was KE = mv2 2 and the values of the momentum and energy depended on the veloity with whih the partile is observed. The differene in this ase is that there is still and energy term even in the ase of zero relative motion. What is this energy? This energy is the energy in the famous formula E = m. We now realize that this formula is not ompletely orret as written. It is more properly written as E v=0 = m. (7.11) For a system that is basially not moving relative to an observer, i. e. a ommover, this is the energy that is neessary to form the system; its rest energy, Setion This will make more sense when we talk about many partile systems, Setion 7.9. In that regard, please note that by dividing Equation 7.10 by Equation 7.7, p = v. (7.12) E For a single partile, with mass m, this is an interesting observation and provides another way to measure the relative veloity of a partile. For the

5 7.4. MASS 125 multi-partile ase, Setion 7.9, it will beome the definition of the system veloity. We should also note that with these formula s for the energy an momentum, we have a new interpretation of. It is a onversion fator from momentum to energy units and even to mass units. For example, for Equation 7.12 to be true p dim = E or, for Equation 7.11, E dim = m. The most ommon way that this onversion is seen is when you see momenta expressed in MeV. MeV is an energy unit, the energy sale of nulear reations. It is 10 6 ev where the ev is the energy that an eletron gains by moving through a voltage of one volt and is the energy sale of hemial reations, see Setion??, or an ev is Joules. A momentum of 1 MeV = Joules = kg m se = kg m se. Sometimes it is even worse than this. The fators of will be suppressed as when you see someone write that the mass of the eletron is 0.51 MeV. What the more areful person means is 0.51 MeV = kg. 7.4 Mass In formulating the appropriate ation for the relativisti partile, we needed to inlude the mass of the partile, see Setion It was indiated that this is the inertial mass, the mass that resists hanges in the state of motion. For a free partile, the more the trajetory deviates from a straight trajetory, the more ation that it osts, the proper time is shorter, and the mass is the weighting fator; the larger the mass, the higher the ation for the same deviation in the trajetory from the straight trajetory. We also saw that to the ommover, the energy of the system is simply related to the mass, E v=0 = m. This mass was interpreted as the energy needed to reate the system, Setion 7.3. This will be better understood when we disuss multi-partile states, Setion 7.9. Note that the following ombination of the energy and momentum does not have the veloity of the partile in it. E 2 p 2 = ( m 2 ) 2 ( m v ) 2 2 (7.13) = (m ) 2 (7.14) Even though E and p have different values depending on the relative motion. No matter what your relative motion to the partile is, the ombination E 2 p 2 is the same and is m 2 4. In fat, sine p and E are the

6 126 CHAPTER 7. RELATIVISTIC DYNAMICS dynamial entities, they are the things that are generally measured in an experiment on elementary partiles. You measure the momentum, p, by seeing how the partile is effeted by a known fore and the energy, E, by a diret energy transfer measurement. This is how the mass of elementary partiles is atually measured. You independently measure E and p and then form the ombination E 2 p 2 to determine the mass. Although mass is not the only identifying harateristi of an elementary partile, it is the most important one. The mirale of this operation is that you disover that with all the experiments that we have performed measuring a tremendous range of p and E that the masses observed are always in a small group of fixed values, the masses of the elementary partiles. All eletrons have a mass of kg, all protons have a mass of kg and so on. This even works for systems that we know are omposite. All arbon twelve nulei, omposed of six protons and six neutron, have a mass of kg, whih is not the mass of the six protons and six neutrons when measured arefully. The are that must be maintained if the mass differene is to be deteted is the reason for the high preision. The mass of a hydrogen atom whih is a proton and an eletron is???? and, again, is not the mass of an proton and an eletron when measured very arefully. We will disuss this problem when we disuss multi-partile states, Setion Kineti Energy of a Single Partile In non-relativisti physis the kineti energy is zero if you are moving with the partile. It is in this sense that we define the kineti energy for a relativisti partile as the energy of motion and thus the energy above the rest energy. KE E E v=0 (7.15) = m{ 1 } 1 (7.16) If v2 1, using (1 + x) n 1 + nx for x 1. KE = m{ 1 1} (7.17) m {1 + v2 1} (7.18) 22

7 7.6. TRANSFORMATIONS OF MOMENTUM AND ENERGY 127 = mv2 2 + (7.19) Thus, in the small v limit, we reover the usual kineti energy of lassial physis. Later, in the setion on appliations, Setion 7.11 Item 1, we will disuss how in interations of partiles, in partiular nulei, energy is onserved, mass is redued, and kineti energy is produed. 7.6 Transformations of Momentum and Energy In Setion 7.4, we disovered that the ombination E2 4 words, the ombination of variables E2 p2 = m 2. In other p2 although both E and p depend 4 on the relative veloity v, does not depend on the relative veloity. Sine the relative veloity is one of the ways to label the Lorentz transformations, the inferene is that the energy and momentum ombine to form a form invariant for the Lorentz transformations. From the definitions of the energy, Equation 7.5, and the definition of the momentum, Equation 7.8, and the fat that the ation, S, was made to be symmetri under the Lorentz transformations, Setion 7.2.1, you an show that the momentum, p, and the energy, atually E, transform like the position, x and the time, t, see Setion?? Equations??: p = p v E (7.20) E = E v p. (7.21) For instane, as we saw in Setion 7.3, a partile moving at a speed v, p = q mv and E = q m2, is seen to be at rest, p = 0 and E = m, by 1 v2 1 v2 an observer moving at v relative 2 to the original observer. In the ative view of transformations, Setion??, we would say that the transformation brings the partile to its rest frame. The same set of ideas an be reversed. To the observer that is at rest with respet to the partile, the energy, E, is m. To an observer moving at a veloity v relative to that observer the partile has momentum p = E q v = q mv 1 v2 1 v2 and energy E = E q 2 1 v2 = q m 1 v2. Remember that this

8 128 CHAPTER 7. RELATIVISTIC DYNAMICS observer sees the partile moving with a veloity of v. In fat, this ould be the another way to derive the addition of veloities formula, Setion In this sense, p and E form a transforming set like x and t. We all anything that transforms like this a four vetor. This nomenlature omes from the fat that in the real world there are three spae oordinates and one time. 7.7 The Energy, Momentum, and Mass of Light When you disuss light in the lassial sense of Maxwell, it is not lear what is meant by the energy and momentum of light as disussed above. For now though, think of light in the modern ontext a partiulate transfer of energy and momentum, Setion??. Note that for a partile the ratio p E = v (7.22) is independent of the mass that we start with and thus holds for all masses inluding zero. For partiles that travel at the speed of light, this implies that the ratio is 1 and that p = E. In other words for these partiles, E 2 p 2 = 0 whih implies that the mass is zero. Partiles that travel at the speed of light are massless and the onverse also holds that massless partiles travel at the speed of light. Note also that massless partiles, for example light, have energy and momentum. Another way to see that is the for any partiulate system that arries energy and momentum, E = m p 2. (7.23) In the limit as m 0, E = p, whih implies that v =. In addition, the momentum and energy of light transform under the Lorentz transformations in the same way as they do for massive partiles, Equations 7.20 and You an easily onvine yourself that for massless partiles, you annot find a transformation the an yield p = 0 in Equation Sine m 2 = E2 p2 is the same for all Lorentz related E and 4 p, if you had p = 0, and m = 0, you will also have E = 0. A massless partile with E = 0 and p = 0 is not there. Conversely, If a light beam, a beam of energy momentum transferred by massless partiles, has energy E, it has momentum p = E and another observer moving relative to the observer that measures that for the beam will measure an E and p given by Equations 7.20 and Notie how my language here has segued into beam energetis regardless of the partiulate nature of the energy.

9 7.8. INTERACTIONS Interations 7.9 Multi-partile Systems 7.10 Rest energy of omposite and elementary systems When you are at rest with respet to a partile the p is zero and E v=0 = m (7.24) This non-zero rest energy is an interesting aspet of speial relativity. As stated above, Setion 7.3, it is this result that is the basis of the statement that there is an equivalene of mass and energy, E = m. Note that you an never bring light to rest and thus it is onsistent to say that light has energy and still meaningless to talk about rest mass Appliations of Energy Momentum 1. If we had redone our ollision problem with relativisti kinematis we would still have found that the energy and momentum are onserved. Said more generally, sine we want all fundamental proesses to be time translation invariant we want energy to be onserved. Prior to relativity we also assumed that mass the amount of stuff was also onserved. When you think about it this is just a prejudie. We do not have a symmetry requiring mass onservation. In other words, you an now think of proesses in whih the mass is hanged. A popular example is D + T He + n MeV. (7.25) This example is popular beause it is the basis for potential ommerial fusion energy. A deuterium nuleus whih is a heavy hydrogen nuleus, one neutron and one proton, and a triton, an even heavier hydrogen nuleus, whih happens to be radioative would serve as fuel for the fusion reator. The D and T are basially at rest. The inoming energy is m D + m T m. The outgoing energy is r He r + mn2. 1 v2 He 1 v2 n

10 130 CHAPTER 7. RELATIVISTIC DYNAMICS Sine there is no momentum oming in the momentum of the He and n are equal and opposite, p He = p n p. Writing the energy in terms of the momentum, Solving for p m D + m T = Looking up the values, m 2 He 4 + p 2 + m 2 n 4 + p 2 (7.26) p = {(m D + m T ) 2 + m 2 He m2 n} 2 4(m D + m T ) 2 m 2 He (7.27) 2. m D = AMU (7.28) m T = AMU (7.29) m He = AMU (7.30) m n = AMU (7.31) These masses are in atomi mass units or AMU and the onversion is 1 AMU = 931 MeV. (7.32) p = AMU This is one ase where you need a alulator. You have to ompute small differenes. Thus the value of p = 163 MeV. Note that the kineti energy of the He and n are: and KE He = m 2 He + p2 m He = AMU (7.33) KE n = m 2 n + p2 m n = AMU (7.34) Thus the KE of the He is 3.56 MeV and the KE of the n is MeV. The total energy that goes into KE or motional energy is MeV