Spacetime and 4vectors 1
Minkowski space = 4dimensional spacetime ≠ Euclidean 4space Each point in Minkowski space is an event. In SR, Minkowski space is an absolute structure (like space in Newtonian theory).
Euclidean 3space: each point has coords (x, y, z); the “metric” is the distance btwn 2 points: dr2 = dx2 + dy2 + dz2 dr2 is invariant under rotation of coords
Minkowski space: each point has coords (x, y, z, ct); the metric is ds2 = c2dt2 – dx2 – dy2 – dz2 ds2 is invariant under LTs; like rotation of coords in Minkowski space! Euclidean space: translations and rotations of coord axes => infinite 2 number of different Cartesian coord systems Minkowski space: translations and rotations of coord axes plus LTs => infinite number of different coord systems (different IFs with Cartesian coord systems)
Review of vectors in Euclidean 3space (“3vectors”) Vectors are absolute quantities, independent of coord system. Representation (x, y, z) depends on coord system chosen.
Prototypical vector: the displacement r Definition of a 3vector: object with 3 components that transforms under a change of coord system (rotations and translations) in exactly the same way as the displacement. Coordinate transformations are 3 accomplished using matrix algebra. e.g.: rotation of coord axes about zaxis:
x' = cos x + sin y r' = R ⋅r y' = x y sin + cos Note that the transformation eqns z' = z are linear. translation: = + r' r r0 Scalar: a number that doesn't depend on coord system In Newtonian theory, time and mass are scalars. Equations involving only vectors and scalars are invariant; i.e, if 4 true in one coord system, then true in all coord systems.
Laws of physics must have this property.
Rotations and translations are linear transformations => sum of 2 vectors is a vector scalar times vector is a vector
Derivative of vector wrt scalar is a vector.
Magnitude a = ∣a∣ of 3vector a = (a , a , a ) : a2 = a 2 + a 2 + a 2 1 2 3 1 2 3 Magnitude is a scalar, since a transforms like r and r2 is invariant.
Scalar product: ⋅ = a b + a b + a b a b 1 1 2 2 3 3 It's a scalar since ∣a + b∣ 2 = (a + b )2 + (a + b )2 + (a + b )2 1 1 2 2 3 3 = a2 + b2 + 2 (a b + a b + a b ) 1 1 2 2 3 3 Vectors in Minkowski space (4vectors) 5
Notation: 3vectors will be lowercase (e.g., r); 4vectors uppercase (R) Prototypical 4vector: displacement R = (x, y, z, ct) Definition of 4vector: an object with 4 components that transforms like R under a Poincaré transformation (includes standard LTs, rotations, translations, and successive applications of these)
Matrix representation of standard LT:
4vectors are absolute entities; components differ in different IFs. An equation involving only 4vectors and scalars that is true in one IF is true in all IFs; called Lorentzinvariant, or covariant, equations. SR: Laws of physics must be Lorentzinvariant. (SR as metatheory) Square of a 4vector = (A , A , A , A ) : 2 = A 2 – A 2 – A 2 – A 2 6 A 1 2 3 4 A 4 1 2 3 A2 can be pos, neg, or 0 and is a scalar, since R2 is invariant. Magnitude A = ∣A2∣1/2 is also obviously a scalar. => scalar product ⋅ = A B – A B – A B – A B is a scalar. A B 4 4 1 1 2 2 3 3 Interval of proper time: d2 ≡ ds2/c2 = dt2 – (dx2 + dy2 + dz2) / c2 d is a scalar and corresponds to time ticked on a clock at rest. For a particle with 3velocity u, dt = (u) d
4velocity:
4acceleration: U = (u, c) A = (' u + a, ' c) (' = d/dt) 7
In instantaneous rest frame (u = 0):
=> A = (a, 0)
Square of 4velocity, evaluated in arbitrary frame:
Evaluating in rest frame yields U2 = c2 immediately. This is a trivial example of the power of relativity: evaluate scalars in the frame that makes the calculation the simplest.
Square of 4accel: A2 = 2 ( = mag of 3accel in rest frame “proper acceleration”) SP 3.13 Rapidity is defined by: cosh = ; sinh = => tanh = 8 Recall: cosh = (e + e) / 2 ; sinh = (e e) / 2 cosh2 sinh2 = 1 => 2 22 = 1 => definition is consistent.
Standard LT: x' = (x ct) = cosh x – sinh ct ct' = (ct x) = sinh x + cosh ct
Note similarity to rotation matrix. Consider 2 boosts along xaxis: followed by , yielding 9 1 2 3 vel add formula:
= + ; rapidity, unlike velocity, adds! 3 1 2
SP 3.4 10 Nomenclature: A 4vector A is ... timelike if A2 > 0 lightlike, or null, if A2 = 0 spacelike if A2 < 0
in analogy with prototype s2 = c2t2 r2
s2 = 0 => light can travel from earlier to later event s2 > 0 => particle moving with v < c can travel from earlier to later event; can transform to a frame where r2 = 0 s2 < 0 => no particle can travel from one event to the other with v ≤ c; can transform to a frame where t = 0
Null and timelike 4vectors are classed together as causal vectors: sign of 4th component is invariant (“futurepointing” if A > 0 4 “pastpointing” if A < 0) 4