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Unformatted text preview: ht cone u༇ Spacelike vectors have posiCve norm o (Dot product with itself) Timelike vectors have negaCve norm. u༇ Lightlike vectors are on the light cone and have zero norm. u༇ 49 Example: Norm of a 4 vector u༇ Dot the 4 vector with itself x = ( ct, x, y, z ) = ( 4, 3, 2, 1)
norm = x ⋅ x = −16 + 9 + 4 + 1 = −2
this is a "future pointing" timelike vector
u༇ Norm>0, spacelike u༇ Norm<0, Cmelike u༇ Norm=0, lightlike 50 QuesCon: Units u༇ To make the laws of physics simple and to make rotaCons in 4D most easily understandable, what should the speed of light be? A)3X108 m/s
B)3X1010 cm/s
C)1 m/s
D)1 cm/s
E)1 51 4 vectors u༇ 4 vectors transform under rotaCons u༇ The dot product of 4 vectors is a scalar and is invariant under rotaCons u༇ The norm of a vector can be posiCve, negaCve or zero o Spacelike, Cmelike or lightlike respecCvely u༇ Transforming to an inerCal frame moving in the x direcCon is a rotaCon in the x t plane. 52 The SpaceCme Interval Δs u༇ The spaceCme interval is the 4D distance between two events. o spacelike if >0 o Cmelike if <0 o invariant under transformaCons (dot product) 53 RotaCons in 4D (Not on tests) RotaCon in the xy plane. β=
γ= All 4 vectors transform the same way. v
c
1
1− β2 cosh θ = γ
tanh θ = β
sinh θ = βγ RotaCon in the xt plane A rotaCon in the xt plane is a “boost” along the x direcCon to another inerCal frame 54 RotaCon Symmetry in 4D u༇ Includes 3D rotaCons o RotaCons in xy, yz, xz planes u༇ Plus symmetry that all the laws of physics are the same in any inerCal frame. o RotaCon in xt plane is boost in the x direcCon o RotaCon in yt plane is boost in the y direcCon o RotaCon in zt plane is boost in the z direcCon u༇ All 4 vectors transform with the save matrix 55 Velocity, Momentum 4 vectors u༇ The spaceCme (posiCon) 4 vector xµ is deﬁned. u༇ To get a velocity 4 vector (or momentum) we cannot diﬀerenCate with respect to Cme. o Cme is a component of a vector! o we need something like a scalar which is invariant under transformaCons. o The proper ;me τ is deﬁned in every frame and thus an invariant. o vx=dx/dτ= γ dx/dt o px=γ m dx/dt u༇ This is the conserved momentum. 56 Compute 4 momentum u༇ Use proper Cme τ to get 4 velocity u༇ 4 velocity square is invariant: check u༇ 4=momentum = mass Cmes 4 velocity u༇ This will give us the Energy equaCon 57 Force EquaCon u༇ Since it is the 4 momentum that is conserved, we will conCnue to deﬁne Force in an inerCal frame to be the Cme derivaCve of the space component of the 4 momentum. u༇ We may also conCnue to deﬁne the 3 velocity and 3 acceleraCon in the frame in terms of the simple Cme derivaCve. 58 Example: Constant Force u༇ An electron is accelerated by a constant Electric ﬁeld. Find the acceleraCon as a funcCon of the electron’s velocity. 59 Example: proton circles in constant B ﬁeld u༇ A proton with momentum p move in a circle in a constant B ﬁeld perpendicular to the moCon. What is the radius of the circle? (the radius of curvature of the proton path) o Work at the point where the proton is at its minimum x, the velocity is in the +y direcCon, and let the B ﬁeld be in the z direcCon. 60 Derive Important...
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 Winter '08
 Hirsch
 Physics

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