GENERAL RELATIVITY S. Erkki Thuneberg. Department of physics. University of Oulu

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1 GENERAL RELATIVITY S Erkki Thuneberg Department of physics University of Oulu 2016

2 Practicalities The web page of the course is The web page contains the exercises and later also the solutions to the exercises. See the web page also for the lecture and exercise times, and possible changes that are made later. There are no lecture notes. The exception is this short summary of special relativity.

3 1. Special relativity The purpose of this section is to discuss special relativity before we go to the general one. Special relativity has been discussed in courses Introduction to relativity 1 and 2. Here we only repeat some essential points, without going to derivations. We consider an inertial coordinate system with cartesian coordinates x, y and z. Time is denoted by t. The four-dimensional space formed by spatial coordinates and the time coordinate is called spacetime. A point (t, x, y, z) of the spacetime is called event. For two events (t 1, x 1, y 1, z 1 ) and (t 2, x 2, y 2, z 2 ) one can define the differences t = t 2 t 1, x = x 2 x 1,.... The corresponding differentials (the limit that the difference goes small) is denoted by dt, dx, dy and dz. Special relativity is based on two postulates: The speed of light c is the same in all inertial coordinate systems.

4 The laws of Nature are the same in all inertial coordinate systems. Newtonian mechanics does not satisfy the first postulate, since in different inertial coordinate systems K and K dr dt dr dt, (1) where r is the distance traversed in K and correspondingly r in K. However, allowing time to be different in the two coordinates, it is possible to satisfy c = dr dt = dr dt. (2) More precisely we should require the following. For light in K we should have c 2 dt 2 dx 2 dy 2 dz 2 = 0. (3) When this holds in K, the corresponding differentials in K should satisfy c 2 dt 2 dx 2 dy 2 dz 2 = 0. (4) Thus for light in the two frames c 2 dt 2 dx 2 dy 2 dz 2 = c 2 dt 2 dx 2 dy 2 dz 2. (5)

5 Now it can be shown that (5) holds more generally for any differences of events, not just those ones satisfying (3). We omit the proof here, but it was shown in the course Introduction to relativity 1. We introduce the notation ds 2 = c 2 dt 2 dx 2 dy 2 dz 2. (6) Based on the above, ds 2 this is independent of the coordinate system where it is calculated. We define x 0 ct, x 1 x, x 2 y and x 3 z. We also define η µν = 1 kun µ = ν = 0, 1 kun µ = ν 0, 0 kun µ ν, where indices µ and ν can take values 0, 1, 2 and 3. This can also be represented as a matrix η µν = Using these we can write (6) as (7). (8) ds 2 = η µν dx µ dx ν = η µν dx µ dx ν. (9) Here we have use the summation rule: whenever the same letter appears in a term both as a

6 superscript and as a subscript, the term has to summed over different values of the index. For example a µ b µ 3 µ=0 a µ b µ. (10) This notation saves us a lot of writing, and only seldom leads to confusion. Another notation in (9) is that we write dx µ instead of (dx ) µ. This notation is to be understood so that the index still is µ (not µ ), the role of the prime on the index is to indicate that this index refers to components in the primed coordinate system. Later on we encounter quantities that have more indices and the primes indicate which of indices refer to the primed coordinate system. As an example, see (13) below. As an exercise of the index notation we write λ µ = (λ 0, λ 1, λ 2, λ 3 ). Then we define λ µ = η µν λ ν = (λ 0, λ 1, λ 2, λ 3 ). We also can define the inner product of λ µ and σ µ = (σ 0, σ 1, σ 2, σ 3 ) by η µν λ µ σ ν = λ µ σ µ = λ µ σ µ = λ 0 σ 0 λ 1 σ 1 λ 2 σ 2 λ 3 σ 3. Let us consider a particle in spacetime moving a along a world line. The time differential dt

7 between its two events depends on the coordinate system. In order to have an invariant quantity, we define proper time τ so that cdτ = ds. Since = we have [ c 2 dτ 2 = c 2 dt 2 dx 2 dy 2 dz 2 ( ) c 2 dx 2 ( ) dy 2 ( ) ] dz 2 dt 2, (11) dt dt dt dτ = 1 v2 dt. (12) c2 If we have v = 0, then dτ = dt. Thus the proper time is the time measured in the coordinate system where the particle is at rest. Let us consider two inertial frames: K and K. The events in these coordinate frames are marked by x µ and x µ. The Lorentz-transformation between the two inertial frames has to be a linear transformation, which we write x µ = Λ µ ν x ν + a µ. (13) The summation over ν here means that this is the

8 same as the matrix expressions x 0 Λ 0 0 Λ 0 1 Λ 0 2 Λ 0 3 x 0 a 0 x 1 Λ x 2 = 1 0 Λ 1 1 Λ 1 2 Λ 1 3 Λ 2 0 Λ 2 1 Λ 2 2 Λ 2 x 1 3 x 2 + a 1 x 3 Λ 3 0 Λ 3 1 Λ 3 2 Λ 3 x 3 a 2. a 3 3 (14) Here the constant term a µ is included, but it can be removed by shifting the origin of the coordinates to coincide. A special case is that the x, y and z axis of K and K are aligned, and K moves with velocity v along the x axis. In this case [Λ µ ν ] = γ v c γ 0 0 v c γ γ , (15) where γ = 1/ 1 v 2 /c 2. Using (13) one can show that this is equivalent to the well known form of the Lorentz transformation x = y = y z = z x vt 1 v 2 /c 2 t = t (v/c2 )x (16) 1 v 2 /c2.

9 For the differential we get from (13) dx µ = Λ µ ν dxν. (17) As argued above, ds 2 = η µν dx µ dx ν should remain invariant. In the primed coordinates d 2 s = η µν dx µ dx ν = η µν Λ µ α dx α Λ ν β dxβ (18) This should be the same as ds 2 = η αβ dx α dx β for all dx α and dx β. This is possible only when η αβ = η µν Λ µ α Λ ν β. (19) This is a general condition that the coordinate transformation has to satisfy. From the Lorentz transformation one can derive the phenomena time dilation and length contraction.

10 ct future of O spacelike events to O past of O x null cone Spacetime diagrams are graphical representations of the x µ space, where one or two of the spatial coordinates are ignored. Typically the time axis is shown as vertical and the space-like axes as horizontal. In these diagrams one can classify the the events with respect to any point as timelike, null and spacelike. With respect to the origin O, the null cone is defined by x 2 + y 2 + z 2 = c 2 t 2. Time like events relative to O are defined by x 2 + y 2 + z 2 < c 2 t 2. These are divided into past and future events (relative to O) as in the figure. Events with x 2 + y 2 + z 2 > c 2 t 2 are called spacelike relative to O. Let us consider a set of four numbers λ µ = (λ 0, λ 1, λ 2, λ 3 ), which is defined in K. We call

11 λ µ as a 4-vector if changing to a different coordinate system, the coordinates change similarly as for the vector dx µ (17): λ µ = Λ µ ν λ ν. (20) 4-vectors are very useful in special (and general) relativity. When a physical law is expressed using four-vectors, it is automatically valid in all inertial frames. The transformation from one frame to another is the same for all four vectors. 4-vectors can be represented in different forms: λ µ = (λ 0, λ 1, λ 2, λ 3 ) = (λ 0, λ). The four vectors are classified as timelike, null and spacelike according to ds 2 = η µν λ µ λ ν being positive, zero, or negative, respectively. Let us consider particle, whose world line is x µ (τ). Here the parameter τ the is the proper time of the particle. We define the four velocity u µ of the particle as the derivative of x µ (τ): u µ = dxµ (21) dτ Here the numerator is a four vector, and the denominator is invariant in coordinate transformation. Thus u µ is a four vector. The

12 four-velocity is the tangent vector to the world line of the particle. u μ ct world line x The 4-momentum of a particle is defined as p µ = mu µ = m dxµ (22) dτ where m is the mass of the particle (more precisely, the rest mass). Because the mass is constant (independent of coordinates), the 4-momentum has to be a 4-vector. Using (12) we can write its components as p µ = m dxµ dt dt dτ = ( cm mv 1 v 2 /c 2, (23) 1 v 2 /c2). Recalling that the energy and the momentum of the particle are E = c 2 m 1 v 2 /c 2, p = mv (24) 1 v 2 /c2,

13 we notice that p µ = ( E c, p). (25) We define the wave 4-vector of a photon k µ = ( 2π λ, k), k = 2π λ ˆn, (26) where λ is the wave length and ˆn the direction of the propagation of the photon. We see that k µ is a null vector, k µ k µ = 0. The 4-momentum of the photon is p µ = h 2π kµ (27) where h is the Planck constant. This also is null vector, in contrast to the 4-momentum of a particle (22), which is always timelike. Comparing to (25), we identify the photon energy E = cp 0 = ch/λ = hν, where ν is the frequency. This is in accordance with our expectation. We write the equation of motion as dp µ dτ = f µ (28) In order to get the equation dp dt = F, (29)

14 the spatial part of f µ has to be equal to γf. One can show that this leads to f µ = ( γf v, γf ). (30) c Electromagnetism The purpose of this section is to show that Maxwell s equations are in good agreement with special relativity. For that, we write Maxwell s equations using 4-vectors and tensors. Maxwell s equations E = ρ ɛ 0 (31) E = B (32) t B = 0 (33) B = E µ 0 ɛ 0 t + µ 0j. (34) where µ 0 ɛ 0 = 1/c 2. It follows from equations (32) and (33) that E and B can be expressed as E = ϕ A (35) t B = A, (36)

15 where ϕ = ϕ(r, t) is the scalar potential and A = A(r, t) the vector potential. It is convenient to define a 4-potential A µ = ( ϕ, A). (37) c Using this we define electromagnetic field tensor F µν = A µ x ν A ν x µ, (38) where A µ = (ϕ/c, A) following the general rule. F µν is a 4-tensor of rank 2. The requirement for this is that it has two indices (and thus 4*4=16 components) and that it transforms with respect to both indices as a 4-vector in Lorentz-transformation, F µ ν = Λ µ α Λ ν β F αβ. (39) 4-vectors can be called as 4-tensors of rank 1 and 4-scalars as 4-tensors of rank 0. Later we will see tensors of higher rank. Here we omit the proof that the construct of the form (38) satisfies the tensor condition (39). This will be proven under more general conditions later in this course.

16 All the 16 components of F µν can straightforwardly be calculated using the equations above. The result is [F µν ] = 0 E x /c E y /c E z /c E x /c 0 B z B y E y /c B z 0 B x E z /c B y B x 0,(40) where E = (E x, E y, E z ) and B = (B x, B y, B z ). For example F 01 = A 0 x 1 A 1 x 0 = (φ/c) x 1 + A1 (ct) = 1 ( ) φ c x 1 + A1 = 1 t c E1 = 1 c E x. (41) By direct calculation one can further show that Maxwell s equations can be written as F µν F µν x ν = µ 0j µ, (42) x σ + F νσ x µ + F σµ = 0. (43) xν where the electric current 4-vector j µ = (ρc, j). In addition we claim that the equation motion (28) coming from the Lorentz force F = q(e + v B) (44)

17 can be written dp µ dτ = qf µν u ν. (45) Hereby we have shown that the equations of classical electromagnetism can be written using 4-vectors and tensors. This quarantees that the laws are valid in arbitrary inertial frame. The purpose of the rest of the course is to show that equations above can also be generalized to be valid in the presence of gravitation, so that classical electromagnetism and gravitation are in good accordance with each other.