Eficacia o no del acuerdo entre Ecuador y Colombia para solucionar este problema

We shall not have occasion to use such bases very often. Mainly, it is important to under- stand that they exist, that not every basis is derivable from a coordinate system. The algebra of coordinate bases is simpler in almost every respect. We may ask why the standard treat- ments of curvilinear coordinates in vector calculus, then, stick to orthonormal bases. The reason is that in such a basis in Euclidean space, the metric has componentsδ_αβ, so the form of the dot product and the equality of vector and one-form components carry over directly from Cartesian coordinates (which have theonlyorthonormal coordinate basis!). In order to gain the simplicity of coordinate bases for vector and tensor calculus, we have to spend time learning the difference between vectors and one-forms!

5.6 L o o k i n g a h e a d

The work we have done in this chapter has developed almost all the notation and concepts we will need in our study of curved spaces and spacetimes. It is particularly important that the student understands §§5.2–5.4because the mathematics of curvature will be developed

139 5.8 Exercises

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by analogy with the development here. What we have to add to all this is a discussion of parallelism, of how to measure the extent to which the Euclidean parallelism axiom fails. This measure is the famous Riemann tensor.

5.7 F u r t h e r r e a d i n g

The Eötvös and Pound–Rebka–Snider experiments, and other experimental fundamentals underpinning GR, are discussed by Dicke (1964), Misner et al.(1973), Shapiro (1980), and Will (1993, 2006). See Hoffmann (1983) for a less mathematical discussion of the motivation for introducing curvature. For an up-to-date review of the GPS system’s use of relativity, see Ashby (2003).

The mathematics of curvilinear coordinates is developed from a variety of points of view in: Abraham and Marsden (1978), Lovelock and Rund (1990), and Schutz (1980b).

5.8 E x e r c i s e s

1 Repeat the argument that led to Eq. (5.1) under more realistic assumptions: suppose a fractionεof the kinetic energy of the mass at the bottom can be converted into a photon and sent back up, the remaining energy staying at ground level in a useful form. Devise a perpetual motion engine if Eq. (5.1) is violated.

2 Explain why auniformexternal gravitational field would raise no tides on Earth. 3 (a) Show that the coordinate transformation (x,y)→(ξ,η) with ξ =x and η=1

violates Eq. (5.6).

(b) Are the following coordinate transformations good ones? Compute the Jacobian and list any points at which the transformations fail.

(i) ξ =(x2+y2)1/2,η=arctan(y/x); (ii) ξ =lnx,η=y;

(iii) ξ =arctan(y/x),η=(x2_+y2₎−1/2_.

4 A curve is defined by{x=f(λ), y=g(λ), 0λ1}. Show that the tangent vector (dx/dλ, dy/dλ) does actually lie tangent to the curve.

5 Sketch the following curves. Which have the same paths? Find also their tangent vectors where the parameter equals zero.

(a) x=sinλ,y=cosλ; (b) x=cos(2πt2),y=sin(2πt2+π); (c) x=s,y=s+4; (d)x=s2,y= −(s−2)(s+2); (e)x=μ,y=1.

6 Justify the pictures in Fig.5.5.

7 Calculate all elements of the transformation matrices α_β and μ_ν for the trans- formation from Cartesian (x,y) – the unprimed indices – to polar (r,θ) – the primed indices.

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8 (a) (Uses the result of Exer. 7.) Letf =x2+y2+2xy, and in Cartesian coordinates

V→(x2_+3y,y2_{+3x),W →(1, 1). Computef as a function ofrandθ, and find} the components ofV andW on the polar basis, expressing them as functions ofr andθ.

(b) Find the components of d˜f in Cartesian coordinates and obtain them in polars (i) by direct calculation in polars, and (ii) by transforming components from Cartesian.

(c) (i) Use the metric tensor in polar coordinates to find the polar components of the one-formsV˜ andW˜ associated withV andW. (ii) Obtain the polar components of

˜

VandW˜ by transformation of their Cartesian components. 9 Draw a diagram similar to Fig.5.6to explain Eq. (5.38). 10 Prove that∇ V, defined in Eq. (5.52), is a1₁tensor.

11 (Uses the result of Exers. 7 and 8.) For the vector field V whose Cartesian compo- nents are (x2+3y,y2+3x), compute: (a) Vα,β in Cartesian; (b) the transformation μ

αβνVα,β to polars; (c) the components Vμ

;ν directly in polars using the Christoffel symbols, Eq. (5.45), in Eq. (5.50); (d) the divergenceVα,αusing your results in (a); (e) the divergenceVμ;μusing your results in either (b) or (c); (f) the divergence Vμ;_μusing Eq. (5.56) directly.

12 For the one-form field p˜ whose Cartesian components are (x2+3y,y2+3x), com- pute: (a)p_α,β in Cartesian; (b) the transformation αμβν pα,β to polars; (c) the components p_μ;ν directly in polars using the Christoffel symbols, Eq. (5.45), in Eq. (5.63).

13 For those who have done both Exers. 11 and 12, show in polars thatg_μαVα;ν =pμ;ν. 14 For the tensor whose polar components are (Arr=r2,Arθ =rsinθ,Aθr =rcosθ,

Aθθ =tanθ), compute Eq. (5.65) in polars for all possible indices.

15 For the vector whose polar components are (Vr =1,Vθ =0), compute in polars all components of the second covariant derivative Vα;μ;ν. (Hint: to find the second derivative, treat the first derivativeVα;μas any

1 1

tensor: Eq. (5.66).) 16 Fill in all the missing steps leading from Eq. (5.74) to Eq. (5.75).

17 Discover how each expressionVβ,αandVμβμαseparately transforms under a change of coordinates (forβ_μα, begin with Eq. (5.44)). Show that neither is the standard tensor law, but that theirsumdoes obey the standard law.

18 Verify Eq. (5.78).

19 Verify that the calculation from Eq. (5.81) to Eq. (5.84), when repeated ford˜randd˜θ, shows them to be a coordinate basis.

20 For a noncoordinate basis{e_μ}, define∇e_μeν− ∇e_νeμ:=cαμνeαand use this in place

of Eq. (5.74) to generalize Eq. (5.75).

21 Consider thex−tplane of an inertial observer in SR. A certain uniformly accelerated observer wishes to set up an orthonormal coordinate system. By Exer. 21, §2.9, his world line is

t(λ)=asinhλ, x(λ)=acoshλ, (5.96) whereais a constant andaλis his proper time (clock time on his wrist watch).

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(a) Show that the spacelike line described by Eq. (5.96) withaas the variable parameter and λ fixed is orthogonal to his world line where they intersect. Changing λ in Eq. (5.96) then generates afamilyof such lines.

(b) Show that Eq. (5.96) defines a transformation from coordinates (t,x) to coordi- nates (λ,a), which form anorthogonalcoordinate system. Draw these coordinates and show that they cover only one half of the original t−xplane. Show that the coordinates are bad on the lines|x| = |t|, so they really cover two disjoint quadrants. (c) Find the metric tensor and all the Christoffel symbols in this coordinate system. This observer will do a perfectly good job, provided that he always uses Christoffel symbols appropriately and sticks to events in his quadrant. In this sense, SR admits accelerated observers. The right-hand quadrant in these coordinates is sometimes calledRindler space, and the boundary linesx= ±tbear some resemblance to the black-hole horizons we will study later.