Physics I Keystone Institute of Technology & Management, Surajgarh Unit V. Postulates of Einstein s special theory of relativity

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1 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V Pstlates f Einstein s speial thery f relativity. Priniple f Physial Eqivalene The laws f physis, (bth ehanis and eletrdynais) epressed in the fr f atheatial eqatins have the sae fr in all the inertial fraes f referenes. This pstlate is the generalizatin fr the wide range f physial phenena and delares that the laws f physis are abslte and niversal.. Cnstany f Speed f Light The speed f light in free spae is abslte. It has the sae vale in all inertial fraes f referenes. That is, speed f light neither depends n relative tin between the sre and the bserver, nr n the diretin f the tin. This law is an eperiental fat. The tw pstlates taen tgether have a nseqene, that n aterial partile an have velity greater than the velity f light in free spae with whatsever energy it is aelerated. Galilean Transfratin The lassial thery ainly invlves the Galilean Transfratin. X v *P vt O O Geetry f Galilean Transfratin at t = 0, O = O After tie t, S travels a distane vt alng.. The diret & reverse transfratins shld be syetrial.. It st be linear in natre. 3. It is nt based n Abslte Paraeters. 4. Transfratin shld be Finite in spae. = vt y = y z = z t = t By: - Manvendra Singh

2 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V where, y, z are the rdinates bserved fr S at tie t, whereas, y, z are the rdinates bserved fr the sae pint fr frae S. Differentiating (..a) V = V v V y = V y V z = V z Hear we bserve the hange in velity in nly diretin. Nw differentiating again a = a a y = a y a z = a z By the abve eqatin we find n hange in ters f aeleratin. Lrentz Transfratin The eqatins an be deded fr the ving r statinary fraes fr nifr relative velity with respet t eah ther. At Origin = y = z = t = 0 = y = z = t = 0 Fr nsideratin f a spherial srfae + y + z t = 0 + y + z t = 0 + y + z t = λ( + y + z t ) λ(0) = fr nsideratin at rigin where v = 0, Hene t = t Fr the linear transfratin let = γ ( - vt) & By: - Manvendra Singh

3 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V = γ ( - vt ) t = γ [t- (-/ γ )/v] By these eqatins we find the Lrentz fatr γ = v Therefre Lrentz transfratin fr spae & tie are as fllws: = vt v y = y z =z t = t v Fr the inverse Lrentz transfr v = + vt v y = y z = z t = t + v v By ptting the vales f, y, z, t in eqatin + y + z t, we finds + y + z t, whih shws the invariane w.r.t. spae. By: - Manvendra Singh 3

4 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V Relativisti law f additin f velities With the help f Lrentz transfratins we an transfr the velity f a partile fr ne frae f referene t anther frae f referene ving with a nstant relative velity with respet t it. Let be the velity f a partile in a prper frae f referene S. The pnents f this velity are d, dy y, dz z dt dt dt Anther frae f referene S ves with a nifr velity v alng X diretin relative t frae S. At instant t, the rdinates f a pint, y, z are given by Lrentz transfratins as vt y y z z v t t Where v Let be the velity f the sae partile in frae S and its pnents are d, dy y, dz z dt dt dt 3 Taing differentials f transfratin eq and sing eq, we get d d vdt v dt 4 dy dy dz dz vd v dt dt dt Dividing eqs 4,5 and 6 by eq 7, we get By: - Manvendra Singh 4

5 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V d v dt v dy y y dt v dz z z dt v Set f eqs 8,9 and 0 is alled as Lrentz transfratin f velities r law f relativity f velities. It is lear fr eqs 9 and 0 that y and z pnents f velity are affeted by the tin in diretin. Siilarly, we an btain y z d v dt v dy y dt v dz z dt v 3 T he if these transfratins are nsistent with the priniple f relativity, we pt eq 8 and in eq in v v d dt v v By: - Manvendra Singh 5

6 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V v v d dt v v It is lear fr the abve reslts that bth the bservers deterine the sae vale f the speed f light, as they st. Einstein s ass-energy eqivalene relatin and eplain its iprtane. Mass-energy eqivalene relatin We nw that arding t pstlates f Einstein s speial thery f relativity, physial laws reain invariant in all inertial fraes f referenes. S des Newtn s send law f tin and wr-energy there. When a partile is sbjeted t a nstant fre, its ineti energy ges n inreasing, t any vale lassially, bt nt relativisti-ally. Bease when velity apprahes velity f light, wr annt add t its velity bt it shall add t ass. Ths, ass and energy shld be inter-nvertible. Arding t wr-energy there: Change in ineti energy = wr dne de F. d Where F is the fre asing displaeent d in its diretin de Fd Fr Newtn s send law f tin d v dv d F v (Relativisti-ally, is nt nstant) dt dt dt Sbstitting the vale f F fr eq in eq dv d de v d dt dt d d de dv vd dt dt de vdv v d 3 We nw that relativisti ass f a partile with velity v is given by where is rest ass f the partile. 4 v By: - Manvendra Singh 6

7 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V By sqaring and rss ltiplying eq 4, we get v 5 Differentiating the abve eq 5, we get d vdv v d d vdv v d 6 Cparing eq 3 and eq 6, we get de d 7 When fre is applied, the bdy aelerates fr rest t velityv, its ass inreases fr t and its ineti energy inreases fr 0 t E. By integrating eq 7, we get E de d 0 Ths, relativisti ineti energy is given by E 8 The ttal energy E f the bdy f rest ass rest energy, therefre is given by the s f ineti energy E and E E E 9 v The abve eq 9 is alled as Einstein s ass-energy relatin. Sine ass and energy are inter-nvertible, their separate nservatin laws d nt eist bt these are gverned by a single law, the law f nservatin f ass and energy. Iprtane Entire nlear energy presses, fissin, fsin and ther nlear reatins are eellent eaples f ass-energy relatin. Pair prdtin where By: - Manvendra Singh 7

8 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V a phtn ( E.MeV ) passing near a heavy nles breas int e and e pair and annihilatin, where psitrn and eletrn bine t prde phtns are ther eaples f energy nverting int ass and ass nverting int energy, respetively. In every heial reatin that evlves energy, a ertain ant f atter disappears, that is, nverted int energy. Cparisn f relativisti and lassial ineti energies f a ving partile Als, fr eq 8 E (Relativisti) v E v Classially, v E E = 0v is lassial epressin fr ineti energy. Relativisti variatin f ass with velity Relativisti variatin f ass with velity Arding t lassial ehanis, the ass f a bdy reains nstant whether at rest r in tin. Bt arding t Einstein s speial thery f relativity ass varies with velity. T deterine the velity dependene f ass, nsider tw inertial fraes S and S in whih frae S is ving with a nstant velity v relative t S alng X diretin. Tw eatly idential bdies A and B, eah f ass, apprah eah ther with eqal speeds (ppsite t eah ther) parallel t -ais, in frae S. By: - Manvendra Singh 8

9 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V Sppse after llisin bth the bdies A and B alese in t ne bdy (in-elasti llisin). Then, arding t law f nservatin f linear ent, ent befre llisin shld be eqal t ent after llisin. Therefre, v 0 v v 0 That is, velity f alesed bdy in frae S will be zer, r it is at rest in S. Nw, nsider the llisin press as it appears in frae S. Applying relativisti law f additin f velities, velity f bdy A and velity f bdy B in frae S will be given by v v v v 3 4 Let s asse that ass f the bdies is variable in frae S and let the be ving with velity and ving with velity. Als, ass f the alesed bdy will be ving with velity v. Then, arding t priniple f nservatin f linear ent, v 5 Sbstitting vales f and fr eqs 3 and 4 in eq 5, we get By: - Manvendra Singh 9

10 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V v v v v v v v v v v v v v Fr eq 3, we get 6 v v v v 7 Siilarly, fr eq 4, we get v v 8 Dividing eq 8 by eq 7, we get v v By: - Manvendra Singh 0

11 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V v v 9 Cparing eq 6 and eq 9, we get 0 Sine bth sides f the eq 0 are independent f eah ther, they will be eqal fr every vale f and if bth sides are eqal t a nstant, say. That is The nstant is alled the rest ass f the bdy whih rrespnds t ass f the bdy at its zer velity. Ths, and In general, if be the ass f the bdy (relativisti ass) when its velity is v, then v Where v Eq sggests that ass is nt invariant and its vale varies with speed f the bdy. That is, a bdy is re assive when ving than at rest. The variatin f ass (ass inreases with velity) is shwn in the figre. By: - Manvendra Singh

12 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V It is lear fr eq that relativisti ass inrease is signifiant nly at speeds apprahing v that f light. Fr sall vales f v, that is, v, is negligible pared t s that. If speed v f the aterial partile bees eqal t velity f light, bees infinite, whih is nt pssible. Hene, it is nt pssible fr a aterial partile t have velity eqal t r greater than velity f light. Speed f light in a edi ving with a velity v by sing Einstein s velity additin frla. Fresnel predited that light wld be dragged alng by ving edia and new wave velity is given by v v n n Where is velity f light in va, v is velity f ving edi and n is refrative inde f the edi. Let v be the velity f light relative t the edi whih itself is ving with velity v and v ' is the velity f light in statinary edi. Arding t Einstein s velity additin frla, the velity f light in ving edi v' v v vv ' By: - Manvendra Singh

13 Physis I Keystne Institte f Tehnlgy & Manageent, Srajgarh Unit V Or v n v v n sine v ' n Or v v v n n 3 Sine v and v', sing binial epansin p t first tw ters, we get v v v n n 4 Or v v v v 5 n n n Or v n n negleting v n 6 The fatr n is alled as light drag effiient whih was eperientally verified by Fizea with water serving as a ving edi. By: - Manvendra Singh 3