Elementary общая Относительность v3.2 Macdonald, Elementary General Relativity v3.2--Macdonald.pdf:

Elementary общая Относительность v3.2 Macdonald

Elementary General Relativity v3.2--Macdonald.pdf

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Date Mar 20, 2005

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The inertial frame postulate for a ﬂat spacetime
An inertial frame can be constructed with any event E as origin, with any orientation, and with any inertial ob ject at E at rest in it. 14...

i.e., if the times in the two directions are equal. A reformulation of the deﬁnition will make it more transparent. Suppose the pulse from Q to P is the reﬂection of the pulse from P to Q. Then tQ = tQ 15...

Lightlike separated events. In this case E and F can be on the worldline of a pulse of light. With c = 1, a universal light speed means c = |∆x/∆t| = 1 , i.e., |∆x| = |∆t|. Since by deﬁnition ∆s = 0 for lightlike separated events, Eq. (1.11) is satisﬁed....

For convenience, let E have coordinates E (0, 0). Fig. 1.10 shows the worldline W of an inertial observer O moving with velocity v = ∆t/∆x in I . (That is not a typo.) L± are the light worldlines through F . Since c = 1 in I , the slope of L± is ±1. Solve simultaneously the equations for L− and W to obtain the coordinates R(∆x, ∆t). Similarly, the equations for L+ and W give S (∆x, ∆t). According to Eq. (1.11), the proper time, as measured by O , between the 1 timelike separated events S and E , and between E and R, is (∆x2 −∆t2 ) 2 . Since the times are equal, E and F are, by deﬁnition, simultaneous in an inertial frame I in which O is at rest. 21...

Suppose that ∆t = 40 hours and the speed of the airplane with respect to the ground is 1000 km/hr. Substitute values to obtain ∆sa − ∆sg = 1.4 × 10−7 s....

2.1 History of Theories of Gravity moon. In 1846, U. LeVerrier, a French mathematician, calculated that a new planet, beyond Uranus, could account for the discrepancy. He wrote J. Galle, an astronomer at the Berlin observatory, telling him where the new planet should be – and Neptune was discovered! It was within 1 arcdegree of LeVerrier’s prediction. Even today, calculations of spacecraft tra jectories are made using Newton’s theory. The incredible accuracy of his theory will be examined further in Sec. 3.3. Nevertheless, Einstein rejected Newton’s theory because it is based on prerelativity ideas about time and space which, as we have seen, are not correct. For example, the acceleration in Eq. (2.1) is instantaneous with respect to a universal time....

2.2 The Key to General Relativity and Moon causes a diﬀerence in their acceleration toward the Sun. The lunar laser experiment shows that this does not happen. This is something that the Dicke and Braginsky experiments cannot test. The last experiment we shall consider as evidence for the three postulates is the terrestrial redshift experiment . It was ﬁrst performed by R. V. Pound and G. A. Rebka in 1960 and then more accurately by Pound and J. L. Snider in 1964. The experimenters put a source of gamma radiation at the bottom of a tower. Radiation received at the top of the tower was redshifted: z = 2.5 × 10−15 , within an experimental error of about 1%. This is a gravitational redshift . According to the discussion following Eq. (1.6), an observer at the top of the tower would see a clock at the bottom run slowly. Clocks at rest at diﬀerent heights in the Earth’s gravity run at diﬀerent rates! Part of the result of the Hafele-Keating experiment is due to this. See Exercise 2.1. We showed in Sec. 1.3 that the assumption Eq. (1.4), necessary for synchronizing clocks at rest in the coordinate lattice of an inertial frame, is equivalent to a zero redshift between the clocks. This assumption fails for clocks at the top and bottom of the tower. Thus clocks at rest in a small coordinate lattice on the ground cannot be (exactly) synchronized. We now show that the experiment provides evidence that clocks at rest in a small inertial lattice can be synchronized. In the experiment, the tower has (upward) acceleration g , the acceleration of Earth’s gravity, in a small inertial lattice falling radially toward Earth. We will show shortly that the same redshift would be observed with a tower having acceleration g in an inertial frame in a ﬂat spacetime. This is another example of small regions of ﬂat and curved spacetimes being alike. Thus it is reasonable to assume that there would be no redshift with a tower at rest in a small inertial lattice in gravity, just as with a tower at rest in an inertial frame. (It is desirable to test this directly by performing the experiment in orbit.) In this way, the experiment provides evidence that the condition Eq. (1.4), necessary for clock synchronization, is valid for clocks at rest in a small inertial lattice. Loosely speaking, we may say that since light behaves “properly” in a small inertial lattice, light accelerates the same as matter in gravity. We now calculate the Doppler redshift for a tower with acceleration g in an inertial frame. Suppose the tower is momentarily at rest when gamma radiation is emitted. The radiation travels a distance h, the height of the tower, in the inertial frame. (We ignore the small distance the tower moves during the ﬂight of the radiation. We shall also ignore the time dilation of clocks in the moving tower and the length contraction – see Appendix 4 – of the tower. These eﬀects are far too small to be detected by the experiment.) Thus the radiation takes time t = h/c to reach the top of the tower. (For clarity we do not take c = 1.) In this time the tower acquires a speed v = g t = g h/c in the inertial frame. From Exercise 1.6, this speed causes a Doppler redshift z= gh v = 2. c c (2.2)...

The global coordinate postulate for a curved spacetime
The events of a curved spacetime can be labeled with coordinates (y 0 , y 1 , y 2 , y 3 ). In the next two sections we give the metric and geodesic postulates of general relativity. We ﬁrst express the postulates in local inertial frames. This local form of the postulates gives them the same physical meaning as in special relativity. We then translate the postulates to global coordinates. This global form of the postulates is unintuitive and complicated but is necessary to carry out calculations in the theory. We can use arbitrary global coordinates in ﬂat as well as curved spacetimes. We can then put the metric and geodesic postulates of special relativity in the same global form that we shall obtain for these postulates for curved spacetimes. We do not usually use arbitrary coordinates in ﬂat spacetimes because inertial frames are so much easier to use. We do not have this luxury in curved spacetimes. It is remarkable that we shall be able to describe curved spacetimes intrinsical ly , i.e., without describing it as curved in a higher dimensional ﬂat space. Gauss created the mathematics necessary to describe curved surfaces intrinsically in 1827. G. B. Riemann generalized Gauss’ mathematics to curved spaces of higher dimension in 1854. His work was extended by several mathematicians. Thus the mathematics necessary to describe curved spacetimes intrinsically was waiting for Einstein when he needed it....