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5.1 On the relation of gravitation to curvature |
Moreover, an object held stationary relative to the ship has ‘weight’ equal to the force required to keep it accelerating with the ship. Just as in ‘real’ gravity, this force is proportional to the mass of the object. A true inertial frame is one which falls freely toward the rear of the ship, at the same acceleration as particles. From this it can be seen that uniform gravitational fields are equivalent to frames that accelerate uniformly relative to inertial frames. This is the principle of equivalence between gravity and acceleration, and is a cornerstone of Einstein’s theory. Although Galileo and Newton would have used different words to describe it, the equivalence principle is one of the foundations of Newtonian gravity.
In more modern terminology, what we have described is called the weak equivalence principle, ‘weak’ because it refers to the way bodies behave only when influenced by gravity. Einstein realized that, in order to create a full theory of gravity, he had to extend this to include the other laws of physics. What we now call the Einstein equivalence principle says that we can discover how all the other forces of nature behave in a gravitational field by postulating that the differential equations that describe the laws of physics have the same local form in a freely falling inertial frame as they do in SR, i.e. when there are no gravitational fields. We shall use this stronger form of the principle of equivalence in Ch. 7.
Before we return to the proof that freely falling frames are inertial, even for photons, we must make two important observations. The first is that our arguments are valid only locally – since the gravitational field of Earth is not uniform, particles some distance away do not remain at uniform velocity in a particular freely falling frame. We shall discuss this in some detail below. The second point is that there are of course an infinity of freely falling frames at any point. They differ in their velocities and in the orientation of their spatial axes, but they all accelerate relative to Earth at the same rate.
The redshift experiment again
Let us now take a different point of view on the Pound–Rebka–Snider experiment. Let us view it in a freely falling frame, which we have seen has at least some of the characteristics of an inertial frame. Let us take the particular frame that is at rest when the photon begins its upward journey and falls freely after that. Since the photon moves a distance h, it takes time t = h to arrive at the top. In this time, the frame has acquired velocity gh downward relative to the experimental apparatus. So the photon’s frequency relative to the freely falling frame can be obtained by the redshift formula
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From Eq. (5.1) we see that if we neglect terms of higher order (as we did to derive Eq. (5.1)), then we get ν(photon emitted at bottom) = ν(in freely falling frame when photon arrives at top). So there is no redshift in a freely falling frame. This gives us a sound basis for postulating that the freely falling frame is an inertial frame.
116 |
Preface to curvature |
Local inertial frames
The above discussion suggests that the gravitational redshift experiment really does not render SR and gravity incompatible. Perhaps we simply have to realize that the frame at rest on Earth is not inertial and the freely falling one – in which there is no redshift and so Fig. 5.2 leads to no contradiction – is the true inertial frame. Unfortunately, this doesn’t completely save SR, for the simple reason that the freely falling frames on different sides of Earth fall in different directions: there is no single global frame which is everywhere freely falling in Earth’s gravitational field and which is still rigid, in that the distances between its coordinate points are constant in time. It is still impossible to construct a global inertial frame, and so the most we can salvage is a local inertial frame, which we now describe.
Consider a freely falling frame in Earth’s gravitational field. An inertial frame in SR fills all of spacetime, but this freely falling frame would not be inertial if it were extended too far horizontally, because then it would not be falling vertically. In Fig. 5.3 the frame is freely falling at B, but at A and C the motion is not along the trajectory of a test particle. Moreover, since the acceleration of gravity changes with height, the frame cannot remain inertial if extended over too large a vertical distance; if it were falling with particles at one height, it would not be at another. Finally, the frame can have only a limited extent in time as well, since, as it falls, both the above limitations become more severe due to the frame’s approaching closer to Earth. All of these limitations are due to nonuniformities in the gravitational field. Insofar as nonuniformities can be neglected, the freely falling frame can be regarded as inertial. Any gravitational field can be regarded as uniform over a small enough region of space and time, and so we can always set up local inertial frames. They are analogous to the MCRFs of fluids: in this case the frame is inertial in only a small region for a small time. How small depends on (a) the strength of the nonuniformities of the gravitational field, and (b) the sensitivity of whatever experiment is being used to detect noninertial properties of the frame. Since any nonuniformity is, in principle, detectable, a frame can only be regarded mathematically as inertial in a vanishingly small region. But for current technology, the freely falling frames near the surface of Earth can be regarded
A
B Earth
C
Figure 5.3 A rigid frame cannot fall freely in the Earth’s field and still remain rigid.
117 |
5.1 On the relation of gravitation to curvature |
as inertial to a high accuracy. We will be more quantitative in a later chapter. For now, we just emphasize the mathematical notion that any theory of gravity must admit local inertial frames: frames that, at a point, are inertial frames of SR.
Tidal forces
Nonuniformities in gravitational fields are called tidal forces, since they are the ones that raise tides. (If Earth were in a uniform gravitational field, it would fall freely and have no tides. Tides bulge due to the difference of the Moon’s and Sun’s gravitational fields across the diameter of Earth.) We have seen that these tidal forces prevent the construction of global inertial frames. It is therefore these forces that are regarded as the fundamental manifestation of gravity in GR.
The role of curvature
The world lines of free particles have been our probe of the possibility of constructing inertial frames. In SR, two such world lines which begin parallel to each other remain parallel, no matter how far they are extended. This is exactly the property that straight lines have in Euclidean geometry. It is natural, therefore, to discuss the geometry of spacetime as defined by the world lines of free particles. In these terms, Minkowski space is a flat space, because it obeys Euclid’s parallelism axiom. It is not a Euclidean space, however, since its metric is different: photons travel on straight world lines of zero proper length. So SR has a flat, nonEuclidean geometry.
Now, in a nonuniform gravitational field, the world lines of two nearby particles which begin parallel do not generally remain parallel. Gravitational spacetime is therefore not flat. In Euclidean geometry, when we drop the parallelism axiom, we get a curved space. For example, the surface of a sphere is curved. Locally straight lines on a sphere extend to great circles, and two great circles always intersect. Nevertheless, sufficiently near to any point, we can pretend that the geometry is flat: the map of a town can be represented on a flat sheet of paper without significant distortion, while a similar attempt for the whole globe fails completely. The sphere is thus locally flat. This is true for all so-called Riemannian2 spaces: they all are locally flat, but the locally straight lines (called geodesics) do not usually remain parallel.
Einstein’s important advance was to see the similarity between Riemannian spaces and gravitational physics. He identified the trajectories of freely falling particles with the geodesics of a curved geometry: they are locally straight since spacetime admits local inertial frames in which those trajectories are straight lines, but globally they do not remain parallel.
2B. Riemann (1826–66) was the first to publish a detailed study of the consequences of dropping Euclid’s parallelism axiom.
118 |
Preface to curvature |
We shall follow Einstein and look for a theory of gravity that uses a curved spacetime to represent the effects of gravity on particles’ trajectories. To do this we shall clearly have to study the mathematics of curvature. The simplest introduction is actually to study curvilinear coordinate systems in a flat space, where our intuition is soundest. We shall see that this will develop nearly all the mathematical concepts we need, and the step to a curved space will be simple. So for the rest of this chapter we will study the Euclidean plane: no more SR (for the time being!) and no more indefinite inner products. What we are after in this chapter is parallelism, not metrics. This approach has the added bonus of giving a more sensible derivation to such often-mysterious formulae as the expression for2 in polar coordinates!
5.2 Te n s o r a l g e b ra i n p o l a r co o rd i n a t e s
Consider the Euclidean plane. The usual coordinates are x and y. Sometimes polar
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which are valid to first order.
It is also possible to use other coordinate systems. Let us denote a general coordinate system by ξ and η:
ξ = ξ (x, y), ξ = ∂∂ξx x + ∂∂ξy
y,
(5.5)
η = η(x, y), η = ∂η x + ∂η y.
∂x ∂y
In order for (ξ , η) to be good coordinates, it is necessary that any two distinct points (x1, y1) and (x2, y2) be assigned different pairs (ξ1, η1) and (ξ2, η2), by Eq. (5.5). For instance, the definitions ξ = x, η = 1 would not give good coordinates, since the distinct points (x = 1, y = 2) and (x = 1, y = 3) both have (ξ = 1, η = 1). Mathematically, this requires that if ξ = η = 0 in Eq. (5.5), then the points must be the same, or x =y = 0. This will be true if the determinant of Eq. (5.5) is nonzero,
det |
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This determinant is called the Jacobian of the coordinate transformation, Eq. (5.5). If the Jacobian vanishes at a point, the transformation is said to be singular there.
119 |
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5.2 Tensor algebra in polar coordinates |
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transformation in the way that the displacement transforms. That is, a vector r can be |
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( ξ , η). Then it is clear that for small ( x, y) we have (from Eq. (5.5)) |
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we can write the transformation for an arbitrary V in the same manner as in SR |
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where unprimed indices refer to (x, y) and primed to (ξ , η), and where indices can only take the values 1 and 2. A vector can be defined as something whose components transform according to Eq. (5.9). There is a more sophisticated and natural way, however. This is the modern way, which we now introduce.
Consider a scalar field φ. Given coordinates (ξ , η) it is always possible to form the
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derivatives ∂φ/∂ξ and ∂φ/∂η. We define the one-form dφ to be the geometrical object whose components are
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and similarly for ∂φ/∂η. The most convenient way to write this in matrix notation is as a transformation on row-vectors,
(∂φ/∂ξ ∂φ/∂η) = |
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because then the transformation matrix for one-forms is defined by analogy with Eq. (5.8) as a set of derivatives of the (x, y)-coordinates by the (ξ , η)-coordinates:
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3We shall denote Euclidean vectors by arrows, and we shall use Greek letters for indices (numbered 1 and 2) to denote the fact that the sum is over all possible (i.e. both) values.
120 |
Preface to curvature |
Using this matrix the component-sum version of the transformation in Eq. (5.12) is
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Note that the summation in this equation is on the first index of the transformation matrix, as we expect when a row-vector pre-multiplies a matrix.
It is interesting that in SR we did not have to worry about row-vectors, because the simple Lorentz transformation matrices we used were symmetric. But if we want to go beyond even the simplest situations, we need to see that one-form components are elements of row-vectors. However, matrix notation becomes awkward when we go beyond tensors with two indices. In GR we need to deal with tensors with four indices, and sometimes even five. As a result, we will normally express transformation equations in their algebraic form, as in Eq. (5.14); students will not see much matrix notation later in this book.
What we have seen in this section is that, in the modern view, the foundation of tensor algebra is the definition of a one-form. This is more natural than the old way, in which a single vector ( x, y) was defined and others were obtained by analogy. Here a whole class of one-forms is defined in terms of derivatives, and the transformation properties of one-forms follow automatically.
Now a vector is defined as a linear function of one-forms into real numbers. The implications of this will be explored in the next paragraph. First we just note that all this is the same as we had in SR, so that vectors do in fact obey the transformation law, Eq. (5.9). It is of interest to see explicitly that ( α β ) and ( α β ) are inverses of each other. The product of the matrices is
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(5.15)
(5.16)
Cu rves and vectors
The usual notion of a curve is of a connected series of points in the plane. This we shall call a path, and reserve the word curve for a parametrized path. That is, we shall follow modern mathematical terminology and define a curve as a mapping of an interval of the real line into a path in the plane. What this means is that a curve is a path with a real number associated with each point on the path. This number is called the parameter s. Each point has coordinates which may then be expressed as a function of s:
121 |
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5.2 Tensor algebra in polar coordinates |
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Curve :{ξ = f (s), η = g(s), a s b} |
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Figure 5.4
defines a curve in the plane. If we were to change the parameter (but not the points) to s = s (s), which is a function of the old s, then we would have
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where f and g are new functions, and where a = s (a), b = s (b). This is, mathematically, a new curve, even though its image (the points of the plane that it passes through) is the same. So there is an infinite number of curves having the same path.
The derivative of a scalar field φ along the curve is dφ/ds. This depends on s, so by changing the parameter we change the derivative. We can write this as
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where V is the vector whose components are (dξ /ds, dη/ds). This vector depends only on
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called the tangent vector. (It clearly lies tangent to curve: see Fig. 5.4.) So a vector may be regarded as a thing which produces dφ/ds, given φ. This leads to the most modern view, that the tangent vector to the curve should be called d/ds. Some relativity texts occasionally
use this notation. For our purposes, however, we shall just let V be the tangent vector whose components are (dξ /ds, dη/ds). Notice that a path in the plane has, at any point, an infinity of tangents, all of them parallel but differing in length. These are to be regarded as vectors tangent to different curves, curves that have different parametrizations in a neighborhood of that point. A curve has a unique tangent, since the path and parameter are given. Moreover, even curves that have identical tangents at a point may not be identical elsewhere. From the
+ ≈ +
Taylor expansion ξ (s 1) ξ (s) dξ /ds, we see that V(s) stretches approximately from s to s + 1 along the curve.
Now, it is clear that under a coordinate transformation s does not change (its definition
had nothing to do with coordinates) but the components of V will, since by the chain rule
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A curve, its parametrization, and its tangent vector.
122 |
Preface to curvature |
To sum up, the modern view is that a vector is a tangent to some curve, and is the
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function that gives dφ/ds when it takes dφ as an argument. Having said this, we are now in a position to do polar coordinates more thoroughly.
Polar coordinate basis one -forms and vectors
The bases of the coordinates are clearly
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The basis one-forms are, analogously, |
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We can draw pictures of the bases at various points (Fig. 5.5). Drawing the basis vectors is no problem. Drawing the basis one-forms is most easily done by drawing surfaces of
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constant r and θ for dr and dθ . These surfaces have different orientations in different places. There is a point of great importance to note here: the bases change from point to point.
For the vectors, the basis vectors at A in Fig. 5.5 are not parallel to those at C. This is because they point in the direction of increasing coordinate, which changes from point to