Mechanics Physics 151

Transcription

1 Mechanics Physics 151 Lecture 15 Special Relativity (Chapter 7)

2 What We Did Last Time Defined Lorentz transformation Linear transformation of 4-vectors that conserve the length in Minkowski space Derived general form of homogeneous Lorentz transformation Product of rotation and proper Lorentz transformation Found explicit matrix expression of PLT HLTs form a group, PLTs don t HLT HLT = HLT PLT PLT = HLT

3 Fun With Paradox In general, two PLTs don t add up to a PLT Rotation becomes involved Example: two objects are moving in parallel rotation! stationary observer Seen from an observer moving upwards moving observer Can you see where the rotation come from?

4 Fun With Paradox How do you know two objects passed the line simultaneously? By sending light and receiving reflection l Light is sent at t = c l l Reflections come back at t = + c l Took same time same distance Came back simultaneously reflected at the same time What happens if the observer was moving?

5 Fun With Paradox For a moving observer A Light is sent at the same moment l l Reflection from A comes back earlier than from B A must have passed earlier! B rotation! Definition of simultaneous depends on the observer Causing this effect

6 What We Did Last Time Discussed Lorentz transformation Linear transformation of 4-vectors that conserve the length in Minkowski space Derived general form of homogeneous Lorentz transformation Product of rotation and proper Lorentz transformation Found explicit matrix expression of PLT HLTs form a group, PLTs don t We got the stage = spacetime Let s get the actors = physical quantities

7 4-Vectors We write 4-vectors as x 0 x = ct 1 x = It seems confusing, but you ll get used to it Let s follow a particle traveling in 4-space Trajectory is given by x 2 x = y λ is a parameter that varies monotonously along the curve Proper time τ is a convenient possibility for λ 3 x At any point on the curve, we can define a tangent 4-vector u dx dτ = = Call it 4-velocity z Greek index = x ( λ), x ( λ), x ( λ), x ( λ)

8 4-Velocity S is observer s frame. S is the particles rest frame Particle s 3-velocity in S is v We define 4-velocity u 0 x = ct =γ cτ 0 0 x = = τ γ c Length is u i i i x = γβ cτ = γvτ 4-velocity is the relativistic extension of v It s timelike, by the way i i x = = γ v τ Roman index = 1 3 Space part is proportional to the 3-velocity 0 0 i i uu uu = γ ( c v) = c Lorentz invariant i

9 4-Momentum Multiplying 4-velocity with mass gives 4-momentum 0 0 p = mu = mγ c Space part is natural extension of 3-momentum Time part is energy This needs to be confirmed after introducing force Lorentz invariance is obvious p p p p = m c 0 0 i i 2 2 Kinetic energy is defined as v= 0 p = mu = mγ v i i i 0 E/ c= p = mγ c T = E E = m c + p c mc E = m c +p c p= mγ v Becomes p 2 /2m at low-speed limit

10 Lorentz Tensor Proper Lorentz transformation turns a 4-vector into another 4-vector Consider it a linear function of 4-vector Express it as x = L x ν ν You can define a whole bunch of quantities using this convention Call them tensors X Upper index = 4-vector Lower index = function that accepts 4-vector X X X ν X ν X ν We ll find their physical meanings as we go

11 Tensor Product Tensor product of two 4-vectors is defined by T u v = Write this as αβ α β T αβ is a tensor of rank 2 T = u v You can repeat this to define tensors of rank n Lorentz transformation of T αβ can be easily found T = u v = L u L v = L L T αβ α β α β ν α β ν ν ν Use as many Lorentz tensor as necessary to convert all indices This (irresponsible-sounding) rule works for Lorentz transformation of any tensor

12 Scalar Product We define the scalar product of two 4-vectors 0 0 i i ν u v= u v u v = u gνv Metric tensor Two lower indices because it takes two 4-vectors and returns a scalar How does it transform? ν α ν β α β u v = u g v = u L g L v = u g v = u v ν α ν β αβ Scalar product is Lorentz invariant, as expected g =

13 Metric Tensor A coordinate system in general has basis vectors u = u e = u e + u e + u e + u e Scalar product u v can be written as u v= u e v ν e g ν = e e ν ν Metric tensor defines the scalar products of the basis vectors Lengths of and angles between the basis vectors That s what metric means

14 General Metric Tensor Metric g ν in Minkowski space is diagonal Coordinate axes are always orthogonal Formalism of tensors allows more flexibility Useful in curved space coordinates General relativity makes full use of this Let s not get into it for now

15 1-Form Let s look at a scalar product in a different way scalar u v= u g v ν ν 4-vector We call u = u g as the 1-form of u ν ν Difference between 4-vector and 1-form seems small Why do we make such distinction? function that turns a 4-vector into a scalar 0 x0 = x = ct non-minkowski metric can make them really different There are physical quantities that are naturally 1-form xi = x i

16 Gradient f( x ) A particle goes along a curve x = x ( τ ) Rate of increase of f( x ( τ )) is given by Consider a scalar function f( x ) dx f( x ) gradient of f = v x dτ x Gradient operates on velocity to make a scalar 1-form Gradient operator is defined by Also known as d But I ll avoid this notation = x Lower index shows it s a 1-form

17 4-Vector and 1-Form 4-vector can be turned into its 1-form by Obviously you can do the reverse ν αβ u = g u where g g ν g ν looks identical to g ν in Minkowski space This gives us Lorentz transformation for 1-form Works just the same as 4-vector, as it should βγ = δ ν ν α ν αβ β u = g u = g L u = g L g u = L u ν ν α ν α β β α γ u = u g ν ν

18 Rank of Tensors A general tensor has n upper and p lower indices Call it a tensor of rank It takes p 4-vectors and n 1-forms and return a scalar αβ γ Tγ aαbβ c = 4-vector and its 1-form are interchangeable using g ν We can turn a tensor into equivalent tensors with different rank, as long as n + p is conserved Example: scalar ( ) n p T a b c = T ( g a ) b c αβ γ αβ δ γ γ α β γ αδ β T αβ ( ) 2 γ is rank 1 β αβ T = T g δγ γ αδ

19 Lorentz Transformation We can find Lorentz transformation for any tensor Transform all indices using Lorentz tensor Example: α T a b β β α = scalar Transform this to get T a b = T L a L b = T a b α β α β ν ν β α β α ν ν We now know all the rules for Lorentz transformation Moving indices up and down They aren t even all that complicated T = L L T α α ν β β ν

20 Force Newton s laws must be correct if the velocity is zero d F= p = p in the rest frame of the object dt Momentum transforms as a 4-vector Time dilation changes the time derivative Natural extension would be K must be a 4-vector dp K dτ = τ is proper time. Connected with t by How do we find the 4-force K? dt = γ dτ

21 Electromagnetic Force We assume Maxwell s equations are always correct Predicts constant speed of light c We want to rewrite it in a covariant form EM force on a charged particle can be derived from the generalized potential U = e( φ A v) 4-velocity u was Define 4-potential as Scalar product is u u = c v 0 (, ) ( γ, γ ) A= A A = c 0 (, ) ( φ, ) i.e. relativistically correct A This should better be a scalar Looks promising Au φ = γ c A γv= γ ( φ A v ) c γ U = ea u

22 Electromagnetic Force U = e( A u ) is a scalar if A = ( φ c, A) is a 4-vector What we are looking for if the laws of EM are covariant Force in 3-d is given by Extend this to 4-d Careful to make it a real 4-vector K F i U d U = + xi dt x i ν ν ( ea u ) d ν ( ea u ) ν A da = = e uν x dτ u x d ν τ Minkowski force for a charged particle in EM field

23 Electromagnetic Force ν 0 i A da A A A A K = e uν = e u0 + ui u0 u i x dτ x x x0 x i Rewrite using E and B E i = φ = c t xi x i 0 i A A A j i i i A A u j = = + xi x j γ ( v B) ( v ( A) ) 0 A bit of work K γ ev E c i i = K = e E + ( v B) i 0 i i γ

24 Electromagnetic Force i dp i i = K = γ e E dτ dp Agrees with e dt = F = E v B Space part ( v B) ( ) i Time part 0 dp dt dp d γ c 0 0 i i K ev E τ = = Rate of work done by the EM force W c = Integrate p = Energy! 0 E c Confirmation promised earlier

25 Faraday Tensor ν ν A da A A K = e uν = e u x dτ x x ν ν ν Tensor F ν is called Faraday, or EM field tensor F ν E and cb form a tensor of rank 2 cf. φ/c and A make a 4-vector ef 0 Ex Ey Ez ν Ex 0 cbz cb A A y = = x x Ey cbz 0 cb ν x Ez cby cbx 0 u ν

26 Other Forces What happens with forces other than EM? There is no general method for making forces covariant You must deal with each force, case by case There are 4 (known) fundamental forces in nature EM, gravity, weak and strong Covariant form has been found for weak and strong Need quantum field theory to do this Gravity cannot be made covariant Need general relativity

27 Summary Defined covariant form of physical quantities 4-vectors: velocity, momentum Tensors: metric, Lorentz 1-forms: gradient Found how to Lorentz transform them Covariant form of Newton s equation with EM force EM potential 4-vector, EM force tensor Equation of motion dp K dτ =