Special relativity - Tran Thi Ngoc Dung

Nội dung text: Special relativity - Tran Thi Ngoc Dung

1. SPECIAL RELATIVITY Albert Einstein (1879-1955) Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016
2. OUTLINES • Einstein’s Postulates • Lorentz Transformation • The Lorentz Velocity Transformation • Relativity of Simultaneity • Relativity of Length –Contraction of length • Relativity of Time interval –Time dilatation • Invariance of Space –Time interval • Relativistic Dynamics • Relativistic Energy, Relativistic Kinetic energy
3. Einstein’s Postulates  Einstein’s first postulate, called the principle of relativity,states:The laws of physics are the same in all inertial frames of reference. dW  Einstein’s second postulatedt states: The speed of light in vacuum is the same in all inertial frames of reference and is independent of the motion of the source
4. The Ultimate Speed Limit  Einstein’s second postulate immediately implies the following result: It is impossible for an inertial observer to travel at c, the speed of light in vacuum.  We can prove this by showing that travel at c implies a logical contradiction. Suppose that the spacecraft is moving at the speed of light c relative to an observer on the earth, so that If the spacecraft turns on a headlight, the second postulate now asserts that the earth observer measures the headlight beam to be also moving at c Thus this observer measures that the headlight beam and the spacecraft move together and are always at the same point in space. But Einstein’s second postulate also asserts that the headlight beam moves at a speed relative to the spacecraft, so they cannot be at the same point in space. This contradictory result can be avoided only if it is impossible for an inertial observer, such as a passenger on the spacecraft, to move at c. As we go through our discussion of relativity, you may find yourself asking the question Einstein asked himself as a 16-year- old student, “What would I see if I were traveling at the speed of light?” Einstein realized only years later that his question’s basic flaw was that he could nottravel at c
5. All observers see light flashes go by them with the same speed v No matter how fast the guy on the rocket is moving!! c Both guys see the light flash travel with velocity = c
6. Even when the light flash is traveling in an opposite direction v c Both guys see the light flash travel past with velocity = c
7. The Lorentz Coordinate Transformation Event A observed in reference frame O at y’ y t’ (x,y,z,t) , is observed at in ereference frame O’(x’,y’,z’,t’) t V=const The relationships between (x,y,z,t) and x’ (x’,y’,z’,t’) are given by the Lorentz O’ Coordinate Transformation O x t=t’=0 , OO’ z z’ v t x x vt 2 x' ; y' y; z' z; t' c v2 v2 1 1 c2 c2 v t' x' x' vt' 2 x ; y y'; z z'; t c v2 v2 1 1 c2 c2
8. Derivation of The Lorentz Coordinate Transformation y’ t=t’=0 , OO’ y t’ t V=const x' (x vt) x (x' vt') ct' (ct vt) x’ ct (ct' vt') O O’ c2tt' 2 (c2 v2 )tt' 1 x  v2 z z’ 1 c2 v ct' (ct x) c v t' (t x) 2 v c t' x' x' vt' 2 x ; y y'; z z'; t c v2 v2 1 1 c2 c2
9. The Lorentz Velocity Transformation  The relationships between The velocity of a point mass in the RF O and RF O’ u(ux ,uy ,uz ),u'(u'x ,u'y ,u'z ) v2 v2 u 1 u 1 u v y 2 z 2 u' x ; u' c ; u' c x v y v z v 1 u x 1 u x 1 u x c2 c2 c2 v2 v2 u' 1 u' 1 u' v y 2 z 2 u x ; u c ; u c x v y v z v 1 u'x 1 u'x 1 u'x c2 c2 c2
10. DERIVATION: u(ux ,uy ,uz ),u'(u'x ,u'y ,u'z ) v dt dx dx vdt 2 dx' ; dy' dy, dz' dz, dt' c v2 v2 1 1 c2 c2 dx vdt 2 v dx 1 v dx' 2 u v c dt u' x dt' v v dx x v dt dx 1 1 u x c2 c2 dt c2 v2 1 c2 v2 dy v2 v2 dy 1 1 u 1 dy' 2 dt 2 y 2 c c u' c dt' v v dx y v dt dx 1 1 u x c2 c2 dt c2 v2 dz v2 v2 dz 1 1 u 1 dz' 2 dt 2 z 2 c c u' c dt' v v dx z v dt dx 1 1 u x c2 c2 dt c2
11. u' v 0.7 0.9 u x c 0.981c x v 1 (0.7 0.9) y’ 1 u' y t’ 2 x c t V=0.9c u x v 0.95 0.9 u'x 0.345c v 1 (0.95*0.9) O’=spaceship 1 u x x’ c2 O=Earth x z z’
12. Relativity of Simultaneity In RF O, two events A and B happen simultaneously :tA=tB In RF O’, two events A and B happen at time, t’A,t’B given by: B occurs v v t A 2 x A t B 2 x B y before A ' c ' c y’ t A t B v v 2 v 2 1 1 c2 c2 v O’ (t B t A ) 2 (x B x A ) O ' ' c x’ t B t A x v 2 xA xB 1 z’ c2 z tA tB ' ' t B t A , x B x A t B t A A occurs v (x x ) before B 2 B A A and B ' ' c ' ' t B t A x B x A t B t A occur v 2 v 1 simultaneou c2 sly O’ ’
13. Relativity of Length –Contraction of length A rod of length lo, at rest in RF O’ The length of the rod in RF O ? You must mark the two ends of the rod at the same time in RF O to measure the Chiều thanh là lo  length. y y’ v2 v   o 1 x’ x’ c2 O’ A B x’ O   o x z z’ x vt ' x A vt A ' x B vt B x' x A ;x B v2 v2 v2 The length l <lo 1 1 1 c2 c2 c2   The proper length is the 2 ' ' x B x A v longest x B x A   o 1  v2 c2  o 1 c2
14. Relativity of Time Intervals –Time Dilation In a particular frame of reference, suppose that two events occur at the same point in space. The time interval between these events, as measured by an observer at rest in this same frame (which we call the rest frame y y’ of this observer), is t Then an observer in a t’=t’ -t’ o v 2 1 second frame moving with constant speed v x’A, relative to the rest frame will measure the time O O’ t x’ v x t' x' z z’ 2 t c ; x' 0 v2 t' 1 t t' c2 v2 1 to c2 t to v2 1 c2
15. How long is the spaceship? A spaceship flies past earth at a speed of 0.990c. A crew member on board the spaceship measures its length, obtaining the value 400 m. What length do observers measure on earth? 2 v 2   o 1 400 1 0.99 56.43m c2
16. Proper Time (Thời gian riêng)  There is only one frame of reference in which a clock is at rest, and there are infinitely many in which it is moving. Therefore the time interval measured between two events (such as two ticks of the clock) that occur at the same point in a particular frame is a more fundamental quantity than the interval between events at different points. We use the term proper time to describe the time interval between two events that occur at the same point.
17. INVARIANCE OF SPACE-TIME INTERVAL – Two events A and B occurs in RF A(x A , yA ,zA ,t A );B(x B , yB ,zB ,t B ) O and O’ . The space-time interval A(x' , y' ,z' ,t' );B(x' , y' ,z' ,t' ) between them in RF O and O’ are: A A A A B B B B s2 c2 t 2 2 s'2 c2 t'2 '2 2 2 2 2 2 2 2 2 2 s c t  c (t B t A ) [(x B x A ) (yB yA ) (zB zA ) ] 2 2 2 2 2 2 2 2 2 s' c t' ' c (t'B t'A ) [(x'B x'A ) (y'B y'A ) (z'B z'A ) ] SPACE-TIME INTERVAL doesn’t depend on the RF. s2 s'2
18. RELATIVISTIC DYNAMICS dp Classical dynamics v<<c, m const,F ma dt Relativistic dynamics: v c, mo m v m dp p mv o v2 F v2 1 1 c2 dt c2
19. RELATIVISTIC ENERGY dp d dW F.ds .ds (mv).ds dt dt dm dv dm dv dW .v m .ds .v m .vdt E m dt dt dt dt 2 2 2 W dE c dm mc moc dW dm 2 dv .v mv Eo mo dt dt dt dm m v dv m d 1 E E mc 2 m c2 E mc 2 o o ( v2 ) o o dt v2 dt v2 dt 2 c2 (1 )3 / 2 c2 (1 )3 / 2 c2 c2 v2 2 m dW o 2 d 1 m d 1 c ( v2 ). o ( v2 ) E mc dt 2 dt 2 2 dt 2 v 3 / 2 v 1/ 2 (1 ) (1 ) c2 c2 v2 v2 m (1 )m dW o 2 2 o d 1 m v dv dm c c ( v2 ) o c2 dt v2 dt 2 v2 dt dt (1 )3 / 2 (1 )3 / 2 c2 c2 W mc2
20. Relativistic Kinetic Energy 2 moc 2 2 1 EK E Eo moc moc 1 v2 v2 1 1 c2 c2 v2 1 v c : E m c2 (1 ) 1/ 2 1 m v2 K o 2 o c 2 1 E m v2 K 2 o
21. Relationship between E and p 2 2 2 2 E p c Eo