| Home / lib / tmp / | ||
Gron O., Hervik S. Einstein's general theory of relativity (book draft, 2004)(538s)_PGr_.pdf |
|
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...
The abstract index notation
One of the most heavily used notation, both in this book and in the physics literature in general, is the abstract index notation. So it is best that we get this sorted out as early as possible. As a general rule, repeated indices means summation! For example, µ αµ βµ αµ βµ ≡...
r − r′ · dS′ . |r − r′ | r − r′ · dS′ |r − r′ |3 d Ω= 4 π, 0, P inside V P outside V...
Figure 1.6: Deformation due to tidal forces In general, tidal forces cause changes of shape....
1.8 The covariance principle
The principle of relativity is a physical principle. It is concerned with physical phenomena. It motivates the introduction of a formal principle called the covariance principle: the equations of a physical theory shall have the same form in every coordinate system. This principle may be fulfilled by every theory by writing the equations in an invariant form. This form is obtained by only using spacetime tensors in the mathematical formulation of the theory. The covariance principle and the equivalence principle may be used to obtain a description of what happens in the presence of gravity. We start with the physical laws as formulated in the special theory of relativity. The laws are then expressed in a covariant way by writing them as tensor equations. They are then valid in an arbitrary accelerated system, but the inertial field (‘fictitious force’) in the accelerated frame is equivalent to a non-vanishing acceleration of gravity. One has thereby obtained a description valid in the presence of a gravitational field (as far as non-tidal effects are concerned). In general, the tensor equations have a coordinate independent form. Yet, such covariant equations need not fulfil the principle of relativity. A physical principle, such as the principle of relativity, is concerned with observable relationships. When one is going to deduce the observable consequences of an equation, one has to establish relations between the tensor-components of the equation and observable physical quantities. Such relations have to be defined, they are not determined by the covariance principle....
1.9 Mach’s principle
Einstein wanted to abandon Newton’s idea of an absolute space. He was attracted by the idea that all motion is relative. This may sound simple, but it leads to some highly non-trivial and fundamental questions. Imagine that the universe consists of only two particles connected by a spring. What will happen if the two particles rotate about each other? Will the string be stretched due to centrifugal forces? Newton would have confirmed that this is indeed what will happen. However, when there is no longer any absolute space that the particles can rotate relatively to, the answer is not as obvious. To observers rotating around stationary particles, the string would not appear to stretch. This situation is, however, kinematically equivalent to the one with rotating particles and observers at rest, which presumably leads to stretching. Such problems led Mach to the view that all motion is relative. The motion of a particle in an empty universe is not defined. All motion is motion relative to something else, i.e. relative to other masses. According to Mach this implies that inertial forces must be due to a particle’s acceleration relative to the great masses of the universe. If there were no such cosmic masses, there would exist no inertial forces. In our string example, if there were no cosmic masses that the particles could rotate relatively to, there would be no stretching of the string. Another example makes use of a turnabout. If we stay on this while it rotates, we feel that the centrifugal force leads us outwards. At the same time we observe that the heavenly bodies rotate. Einstein was impressed by Mach’s arguments, which likely influenced Einstein’s construction of the general theory of relativity. Yet it is clear that general relativity does not fulfil all requirements set by Mach’s principle. There exist, for example, general relativistic, rotating cosmological models, where free particles will tend to rotate relative to the cosmic mass of the model....
2.5 The relativity of simultaneity
Events that happen at the same point of time are said to be simultaneous events. We shall now show that according to the special theory of relativity, events that are simultaneous in one reference frame are not simultaneous in another reference frame moving with respect to the first. This is what is meant by the expression “the relativity of simultaneity”....
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)...
2.8 Lorentz-invariant interval
Let two events be given. The coordinates of the events, as referred to two different reference frames Σ and Σ′ are connected by a Lorentz transformation. The coordinate differences are therefore connected by ∆t = γ (∆t′ + ∆y = ∆y ′ ,
v ′ c2 ∆x ),...
where dτ is the proper-time interval between the events, measured with a clock following the curve. The spacetime interval between two events is given by the integral (2.46). It follows that the proper-time interval between two events is path dependent. This leads to the following surprising result: A time-like interval between two events is greatest along the straightest possible curve between them....
However, the relation between the magnetic and the electric force was not fully understood until Einstein had constructed the special theory of relativity. Only then could one clearly see the relationship between the magnetic force on a charge moving near a current carrying wire and the electric force between charges. We shall consider a simple model of a current carrying wire in which we assume that the positive ions are at rest while the conducting electrons move with the velocity v . The charge per unit length for each type of charged parˆ ticle is λ = S ne where S is the cross-sectional area of the wire, n the number of particles of one type per unit length and e the charge of one particle. The current in the wire is ˆ J = S nev = λv . (2.78) ˆ ˆ The wire is at rest in an inertial frame Σ. As observed in Σ it is electrically neutral. Let a charge q move with a velocity u along the wire in the opposite direction of the electrons. The rest frame of q is Σ. The wire will now be described from Σ (see Fig. 2.16 and 2.17).
v...
Problems (b) Two successive Lorentz transformations are given by the matrix product ¯ of each matrix. Find Λµ Λα ′ and Λµ Λα ′ . Are the product of two boosts αµ αµ a boost? The matrix for a general boost in arbitrary direction is given by Λ00 Λ0m Λm ′ m γ = γ, Λm = γ βm , 0 βm βm′ m (γ − 1), = δ m′ + β2 1 1 = , β 2 = β m βm , − β2 =...
| © 2007 eKnigu | ||