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Gron O., Hervik S. Einstein's general theory of relativity (book draft, 2004)(538s)_PGr_.pdf |
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Size 4.0Mb Date Oct 29, 2005 |
“Paradoxically, physicists claim that gravity is the weakest of the fundamental forces.”
Prof. Hallstein Høgåsen– after having fallen from a ladder and breaking both his arms...
Whenever we write the indices between two vertical lines, we mean that the indices shall be well ordered. For a set, µ1 µ2 ...µp , to be well ordered means that µ1 ≤ µ2 ≤ ... ≤ µp . Thus an expression like, Tµν S |µν | means that we shall only sum over indices where µ ≤ ν . We usually use this notation when S |µν | is antisymmetric, which avoids the over-counting of the linearly dependent components. The following notation is also convenient to get straight right away. Here, Aµ...ν is an arbitrary tensor (it may have indices upstairs as well). eα (Aµ...ν ) = Aµ...ν,α ∇α Aµ...ν = Aµ...ν ;α £X d d† ⋆ ⊗ ∧ Partial derivative Covariant derivative Lie derivative with respect to X Exterior derivative operator Codifferential operator Hodge’s star operator Covariant Laplacian Tensor product Wedge product, or exterior product...
1.3 The principle of Relativity
At the beginning of this century Einstein realised that Newton’s absolute space is a concept without physical content. This concept should therefore be removed from the description of the physical world. This conclusion is in accordance with the negative result of the Michelson–Morley experiment [MM87]. In this experiment one did not succeed in measuring the velocity of the Earth through the so-called ‘ether ’, which was thought of as a ‘materialization’ of Newton’s absolute space....
Note that the ∇ operator acts on the coordinates of the field point, not of the source point. Calculating φ(r) from Eq. (1.12) it will be useful to introduce Einstein’s summation convention. For arbitrary a and b one has aj bj ≡ jn aj bj (1.14)...
Eqs. (1.41) and (1.42) have among others the following consequence. If an elastic circular ring is falling freely in the gravitational field of the Earth, as shown in Fig. 1.6, it will be stretched in the vertical direction and compressed in the horizontal direction....
Problems cases. (Consider the head and feet as point particles, each weighing 5kg.) 1. The human is standing on the surface of a Black Hole with 10 times the Solar mass. 2. On the Sun’s surface. 3. On the Earth’s surface. 1.7. Non-relativistic Kepler orbits (a) Consider first the Newtonian gravitational potential ϕ(r) at a distance r M from the Sun to be ϕ(r) = − Gr , where M is the solar mass. Write down the classical Lagrangian in spherical coordinates (r, θ, φ) for a planet with mass m. The Sun is assumed to be stationary. What is the physical interpretation of the canonical momentum pφ = ℓ? How can we from the Lagrangian see that it is a constant of motion? Find the Euler-Lagrange equation for θ and show that it can be written = m d ℓ2 ˙ 0. (1.59) r4 θ2 + dt m sin2 θ Show, using this equation, that the planet can be considered to move in a ˙ plane such that at t = 0, θ = π /2 and θ = 0. (b) Find the Euler-Lagrange equation for r and use it to find r as a function of φ. Show that the bound orbits are ellipses. Of circular orbits, what is the orbital period T in terms of the radius R? (c) If the Sun is not completely spherical, but slightly squashed at the poles, then the gravitational potential along the equatorial plane has to be modified to ϕ(r) = − GM Q − 3, r r (1.60)...
2.1 Coordinate systems and Minkowski-diagrams
The most simple physical phenomenon that we can describe is called an event. This is an incident that takes place at a certain point in space and at a certain point in time. A typical example is the flash from a flashbulb. A complete description of an event is obtained by giving the position of the event in space and time. Assume that our observations are made with reference to a reference frame. We introduce a coordinate system into our reference frame. Usually it is advantageous to employ a Cartesian coordinate system. This may be thought of as a cubic lattice constructed by measuring rods. If one lattice point is chosen as origin, with all coordinates equal to zero, then any other lattice point has three spatial coordinates equal to the distance of that point from the coordinate axes that pass through the origin. The spatial coordinates of an event are the three coordinates of the lattice point at which the event happens....
For the light moving from A to B we may use Eq. (2.18), and for the light from B to A Eq. (2.17). This gives ∆tB = L L 2L + = γ2 . c−v c+v c (2.21)...
is called a line-element. The physical interpretation of the line-element between two infinitesimally close events on a time-like curve is ds2 = −c2 dτ 2 , (2.48)...
2.10 Hyperbolic motion
With reference to an inertial reference frame it is easy to describe relativistic accelerated motion. The special theory of relativity is in no way limited to describe motion with constant velocity. Let a particle move with a variable velocity u(t) = dx/dt along the x-axis in Σ. The frame Σ′ moves with velocity v in the same direction relative to Σ. In this frame the particle-velocity is u′ (t′ ) = dx′ /dt′ . At every moment the velocities u and u′ are connected by the relativistic formula for velocity addition, Eq. (2.33). Thus, a velocity change du′ in Σ′ and the corresponding velocity change du in Σ are related – using Eq. (2.30) – by dt′ + 1 dt =
v ′ c2 dx...
2.13 Tachyons
Particles cannot pass the velocity-barrier represented by the velocity of light. However, the special theory of relativity permits the existence of particles that have always moved with a velocity v > c. Such particles are called tachyons [Rec78]. Tachyons have special properties that have been used in the experimental searches for them. There is currently no observational evidence for the physical existence of tachyons [Kre73]. There are also certain theoretical difficulties with the existence of tachyons. The special theory of relativity, applied to tachyons leads to the following paradox. Using a tachyon telephone a person, A, emits a tachyon to B at a point of time t1 . B moves away from A. The tachyon is reflected by B and reach A before it was emitted, see Fig.2.15. If the tachyon could carry information it might bring an order to destroy the tachyon emitter when it arrives back at A.
ct ct′...
2.14 Magnetism as a relativistic second-order effect
Electricity and magnetism are described completely by Maxwell’s equations of the electromagnetic field, 1 ρq ε0 ∇·B = 0 ∂B ∇×E = − ∂t ∇·E = ∇ × B = µ0 j + together with Lorentz’s force-law F = q (E + v × B). (2.77) 1 ∂E c2 ∂ t (2.73) (2.74) (2.75) (2.76)...
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