- •Contents
- •Preface to the second edition
- •Preface to the first edition
- •1 Special relativity
- •1.2 Definition of an inertial observer in SR
- •1.4 Spacetime diagrams
- •1.6 Invariance of the interval
- •1.8 Particularly important results
- •Time dilation
- •Lorentz contraction
- •Conventions
- •Failure of relativity?
- •1.9 The Lorentz transformation
- •1.11 Paradoxes and physical intuition
- •The problem
- •Brief solution
- •2 Vector analysis in special relativity
- •Transformation of basis vectors
- •Inverse transformations
- •2.3 The four-velocity
- •2.4 The four-momentum
- •Conservation of four-momentum
- •Scalar product of two vectors
- •Four-velocity and acceleration as derivatives
- •Energy and momentum
- •2.7 Photons
- •No four-velocity
- •Four-momentum
- •Zero rest-mass particles
- •3 Tensor analysis in special relativity
- •Components of a tensor
- •General properties
- •Notation for derivatives
- •Components
- •Symmetries
- •Circular reasoning?
- •Mixed components of metric
- •Metric and nonmetric vector algebras
- •3.10 Exercises
- •4 Perfect fluids in special relativity
- •The number density n
- •The flux across a surface
- •Number density as a timelike flux
- •The flux across the surface
- •4.4 Dust again: the stress–energy tensor
- •Energy density
- •4.5 General fluids
- •Definition of macroscopic quantities
- •First law of thermodynamics
- •The general stress–energy tensor
- •The spatial components of T, T ij
- •Conservation of energy–momentum
- •Conservation of particles
- •No heat conduction
- •No viscosity
- •Form of T
- •The conservation laws
- •4.8 Gauss’ law
- •4.10 Exercises
- •5 Preface to curvature
- •The gravitational redshift experiment
- •Nonexistence of a Lorentz frame at rest on Earth
- •The principle of equivalence
- •The redshift experiment again
- •Local inertial frames
- •Tidal forces
- •The role of curvature
- •Metric tensor
- •5.3 Tensor calculus in polar coordinates
- •Derivatives of basis vectors
- •Derivatives of general vectors
- •The covariant derivative
- •Divergence and Laplacian
- •5.4 Christoffel symbols and the metric
- •Calculating the Christoffel symbols from the metric
- •5.5 Noncoordinate bases
- •Polar coordinate basis
- •Polar unit basis
- •General remarks on noncoordinate bases
- •Noncoordinate bases in this book
- •5.8 Exercises
- •6 Curved manifolds
- •Differential structure
- •Proof of the local-flatness theorem
- •Geodesics
- •6.5 The curvature tensor
- •Geodesic deviation
- •The Ricci tensor
- •The Einstein tensor
- •6.7 Curvature in perspective
- •7 Physics in a curved spacetime
- •7.2 Physics in slightly curved spacetimes
- •7.3 Curved intuition
- •7.6 Exercises
- •8 The Einstein field equations
- •Geometrized units
- •8.2 Einstein’s equations
- •8.3 Einstein’s equations for weak gravitational fields
- •Nearly Lorentz coordinate systems
- •Gauge transformations
- •Riemann tensor
- •Weak-field Einstein equations
- •Newtonian limit
- •The far field of stationary relativistic sources
- •Definition of the mass of a relativistic body
- •8.5 Further reading
- •9 Gravitational radiation
- •The effect of waves on free particles
- •Measuring the stretching of space
- •Polarization of gravitational waves
- •An exact plane wave
- •9.2 The detection of gravitational waves
- •General considerations
- •Measuring distances with light
- •Beam detectors
- •Interferometer observations
- •9.3 The generation of gravitational waves
- •Simple estimates
- •Slow motion wave generation
- •Exact solution of the wave equation
- •Preview
- •Energy lost by a radiating system
- •Overview
- •Binary systems
- •Spinning neutron stars
- •9.6 Further reading
- •10 Spherical solutions for stars
- •The metric
- •Physical interpretation of metric terms
- •The Einstein tensor
- •Equation of state
- •Equations of motion
- •Einstein equations
- •Schwarzschild metric
- •Generality of the metric
- •10.5 The interior structure of the star
- •The structure of Newtonian stars
- •Buchdahl’s interior solution
- •10.7 Realistic stars and gravitational collapse
- •Buchdahl’s theorem
- •Quantum mechanical pressure
- •White dwarfs
- •Neutron stars
- •10.9 Exercises
- •11 Schwarzschild geometry and black holes
- •Black holes in Newtonian gravity
- •Conserved quantities
- •Perihelion shift
- •Post-Newtonian gravity
- •Gravitational deflection of light
- •Gravitational lensing
- •Coordinate singularities
- •Inside r = 2M
- •Coordinate systems
- •Kruskal–Szekeres coordinates
- •Formation of black holes in general
- •General properties of black holes
- •Kerr black hole
- •Dragging of inertial frames
- •Ergoregion
- •The Kerr horizon
- •Equatorial photon motion in the Kerr metric
- •The Penrose process
- •Supermassive black holes
- •Dynamical black holes
- •11.6 Further reading
- •12 Cosmology
- •The universe in the large
- •The cosmological arena
- •12.2 Cosmological kinematics: observing the expanding universe
- •Homogeneity and isotropy of the universe
- •Models of the universe: the cosmological principle
- •Cosmological metrics
- •Cosmological redshift as a distance measure
- •The universe is accelerating!
- •12.3 Cosmological dynamics: understanding the expanding universe
- •Critical density and the parameters of our universe
- •12.4 Physical cosmology: the evolution of the universe we observe
- •Dark matter and galaxy formation: the universe after decoupling
- •The early universe: fundamental physics meets cosmology
- •12.5 Further reading
- •Appendix A Summary of linear algebra
- •Vector space
- •References
- •Index
6 |
Curved manifolds |
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6.1 D i f f e re n t i a b l e m a n i f o l d s a n d t e n s o r s
The mathematical concept of a curved space begins (but does not end) with the idea of a manifold. A manifold is essentially a continuous space which looks locally like Euclidean space. To the concept of a manifold is added the idea of curvature itself. The introduction of curvature into a manifold will be the subject of subsequent sections. First we study the idea of a manifold, which we can regard as just a fancy word for ‘space’.
M anifolds
The surface of a sphere is a manifold. So is any m-dimensional ‘hyperplane’ in an n- dimensional Euclidean space (m n). More abstractly, the set of all rigid rotations of Cartesian coordinates in three-dimensional Euclidean space will be shown below to be a manifold. Basically, a manifold is any set that can be continuously parametrized. The number of independent parameters is the dimension of the manifold, and the parameters themselves are the coordinates of the manifold. Consider the examples just mentioned. The surface of a sphere is ‘parametrized’ by two coordinates θ and φ. The m-dimensional ‘hyperplane’ has m Cartesian coordinates, and the set of all rotations can be parametrized by the three ‘Euler angles’, which in effect give the direction of the axis of rotation (two parameters for this) and the amount of rotation (one parameter). So the set of rotations is a manifold: each point is a particular rotation, and the coordinates are the three parameters. It is a three-dimensional manifold. Mathematically, the association of points with the values of their parameters can be thought of as a mapping of points of a manifold into points of the Euclidean space of the correct dimension. This is the meaning of the fact that a manifold looks locally like Euclidean space: it is ‘smooth’ and has a certain number of dimensions. It must be stressed that the large-scale topology of a manifold may be very different from Euclidean space: the surface of a torus is not Euclidean, even topologically. But locally the correspondence is good: a small patch of the surface of a torus can be mapped 1–1 into the plane tangent to it. This is the way to think of a manifold: it is a space with coordinates, that locally looks Euclidean but that globally can warp, bend, and do almost anything (as long as it stays continuous).
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6.1 Differentiable manifolds and tensors |
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Differential structure |
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We shall really only consider ‘differentiable manifolds’. These are spaces that are con- |
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tinuous and differentiable. Roughly, this means that in the neighborhood of each point |
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in the manifold it is possible to define a smooth map to Euclidean space that preserves |
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derivatives of scalar functions at that point. The surface of a sphere is differentiable |
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everywhere. That of a cone is differentiable except at its apex. Nearly all manifolds |
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of use in physics are differentiable almost everywhere. The curved spacetimes of GR |
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certainly are. |
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The assumption of differentiability immediately means that we can define one-forms |
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and vectors. That is, in a certain coordinate system on the manifold, the members of the set |
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dφ; and any set of the form |
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a and b are functions, is also a one-form field. Similarly, every curve (with parameter, say, |
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dφ into the |
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λ) has a tangent vector V defined as the linear function that takes the one-form ˜ |
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derivative of φ along the curve, dφ/dλ: |
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Any linear combination of vectors is also a vector. Using the vectors and one-forms so |
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defined, we can build up the whole set of tensors of type |
M , just as we did in SR. Since |
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we have not yet picked out any |
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tensor to serve as |
the metric, there is not yet any |
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correspondence between forms |
and vectors. Everything else, however, is exactly as we had |
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in SR and in polar coordinates. All of this comes only from differentiability, so the set of all tensors is said to be part of the ‘differential structure’ of the manifold. We will not have much occasion to use that term.
Review
It is useful here to review the fundamentals of tensor algebra. We can summarize the following rules.
(1)A tensor field defines a tensor at every point.
(2)Vectors and one-forms are linear operators on each other, producing real numbers. The linearity means:
˜ p, aV
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where a and b are any scalar fields.
(3)Tensors are similarly linear operators on one-forms and vectors, producing real numbers.
(4)If two tensors of the same type have equal components in a given basis, they have equal components in all bases and are said to be identical (or equal, or the same). Only tensors of the same type can be equal. In particular, if a tensor’s components are all zero in one basis, they are zero in all, and the tensor is said to be zero.
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Curved manifolds |
(5)A number of manipulations of components of tensor fields are called ‘permissible tensor operations’ because they produce components of new tensors:
(i)Multiplication by a scalar field produces components of a new tensor of the same type.
(ii)Addition of components of two tensors of the same type gives components of a new tensor of the same type. (In particular, only tensors of the same type can be added.)
(iii)Multiplication of components of two tensors of arbitrary type gives components of a new tensor of the sum of the types, the outer product of the two tensors.
(iv)Covariant differentiation (to be discussed later) of the components of a tensor of
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type M gives components of a tensor of type |
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(v)Contraction on a pair of indices of the components of a tensor of type MN pro-
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M 1
an upper and lower index.)
(6)If an equation is formed using components of tensors combined only by the permissible tensor operations, and if the equation is true in one basis, then it is true in any other. This is a very useful result. It comes from the fact that the equation (from (5) above) is simply an equality between components of two tensors of the same type, which (from (4)) is then true in any system.
6.2 R i e m a n n i a n m a n i f o l d s
So far we have not introduced a metric on to the manifold. Indeed, on certain manifolds a metric would be unnecessary or inconvenient for whichever problem is being considered. But in our case the metric is absolutely fundamental, since it will carry the information about the rates at which clocks run and the distances between points, just as it does in
SR. A differentiable manifold on which a symmetric |
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Riemannian manifold. (Strictly speak- |
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0 – is it |
called Riemannian; indefinite metrics, like SR and GR, are called pseudo-Riemannian. This is a distinction that we won’t bother to make.) It is important to understand that in picking out a metric we ‘add’ structure to the manifold; we shall see that the metric completely defines the curvature of the manifold. Thus, by our choosing one metric g the manifold gets a certain curvature (perhaps that of a sphere), while a different g would give it a different curvature (perhaps an ellipsoid of revolution). The differentiable manifold itself is ‘primitive’: an amorphous collection of points, arranged locally like the points of Euclidean space, but not having any distance relation or shape specified. Giving the metric g gives it a specific shape, as we shall see. From now on we shall study Riemannian manifolds, on which a metric g is assumed to be defined at every point.
(For completeness we should remark that it is in fact possible to define the notion of curvature on a manifold without introducing a metric (so-called ‘affine’ manifolds). Some
145 |
6.2 Riemannian manifolds |
texts actually approach the subject this way. But since the metric is essential in GR, we shall simply study those manifolds whose curvature is defined by a metric.)
The met ric and local flatness
The metric, of course, provides a mapping between vectors and one-forms at every point.
P
Thus, given a vector field V( ) (which notation means that V depends on the posi-
P P ˜ P = P
tion , where is any point), there is a unique one-form field V( ) g(V( ), ).
˜ P P
The mapping must be invertible, so that associated with V( ) there is a unique V( ). The components of g are called gαβ ; the components of the inverse matrix are called gαβ . The metric permits raising and lowering of indices in the same way as in SR, which means
Vα = gαβ Vβ .
In general, {gαβ } will be complicated functions of position, so it will not be true that there would be a simple relation between, say, V0 and V0 in an arbitrary coordinate system.
Since we wish to study general curved manifolds, we have to allow any coordinate system. In SR we only studied Lorentz (inertial) frames because they were simple. But because gravity prevents such frames from being global, we shall have to allow all coordinates, and hence all coordinate transformations, that are nonsingular. (Nonsingular means, as in § 5.2, that the matrix of the transformation, α β ≡ ∂xα /∂xβ , has an inverse.) Now, the matrix (gαβ ) is a symmetric matrix by definition. It is a well-known theorem of matrix algebra (see Exer. 3, § 6.9) that a transformation matrix can always be found that will make any symmetric matrix into a diagonal matrix with each entry on the main diagonal either +1, −1, or zero. The number of +1 entries equals the number of positive eigenvalues of (gαβ ), while the number of −1 entries is the number of negative eigenvalues. So if we choose g originally to have three positive eigenvalues and one negative, then we can always find a α β to make the metric components become
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≡ (ηαβ ). (6.2)
From now on we will use ηαβ to denote only the matrix in Eq. (6.2), which is of course the metric of SR.
There are two remarks that must be made here. The first is that Eq. (6.2) is only possible if we choose (gαβ ) from among the matrices that have three positive and one negative eigenvalues. The sum of the diagonal elements in Eq. (6.2) is called the signature of the metric. For SR and GR it is +2. Thus, the fact that we have previously deduced from physical arguments that we can always construct a local inertial frame at any event, finds its mathematical representation in Eq. (6.2), that the metric can be transformed into ηαβ at
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Curved manifolds |
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that point. This in turn implies that the metric has to have signature +2 if it is to describe |
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The second remark is that the matrix α β that produces Eq. (6.2) at every point may |
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transformation only if Eq. (5.95) holds: |
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In a general gravitational field this will be impossible, because otherwise it would imply the existence of coordinates for which Eq. (6.2) is true everywhere: a global Lorentz frame. However, having found a basis at a particular point P for which Eq. (6.2) is true, it is possible to find coordinates such that, in the neighborhood of P, Eq. (6.2) is ‘nearly’ true. This is embodied in the following theorem, whose (rather long) proof is at the end of this section. Choose any point P of the manifold. A coordinate system {xα } can be found whose origin is at P and in which:
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That is, the metric near P is approximately that of SR, differences being of second order in the coordinates. From now on we shall refer to such coordinate systems as ‘local Lorentz frames’ or ‘local inertial frames’. Eq. (6.3) can be rephrased in a somewhat more precise way as:
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but generally
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for at least some values of α, β, γ , and μ if the manifold is not exactly flat.
The existence of local Lorentz frames is merely the statement that any curved space has a flat space ‘tangent’ to it at any point. Recall that straight lines in flat spacetime are the world lines of free particles; the absence of first-derivative terms (Eq. (6.5)) in the metric of a curved spacetime will mean that free particles are moving on lines that are locally straight in this coordinate system. This makes such coordinates very useful for us, since the equations of physics will be nearly as simple in them as in flat spacetime, and if constructed by the rules of § 6.1 will be valid in any coordinate system. The proof of this theorem is at the end of this section, and is worth studying.
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6.2 Riemannian manifolds |
Le ngths and volumes
The metric of course gives a way to define lengths of curves. Let dx be a small vector displacement on some curve. Then dx has squared length ds2 = gαβ dxα dxβ . (Recall that we call this the line element of the metric.) If we take the absolute value of this and take its square root, we get a measure of length: dl ≡ |gαβ dxα dxβ |1/2. Then integrating it gives
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where λ is the parameter of the curve* (whose endpoints* |
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as the length of the arbitrary curve.
The computation of volumes is very important for integration in spacetime. Here, we mean by ‘volume’ the four-dimensional volume element we used for integrations in Gauss’ law in § 4.4. Let us go to a local Lorentz frame, where we know that a small fourdimensional region has four-volume dx0 dx1 dx2 dx3, where {xα } are the coordinates which at this point give the nearly Lorentz metric, Eq. (6.3). In any other coordinate system {xα } it is a well-known result of the calculus of several variables that:
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This would be a rather tedious way to calculate the Jacobian, but there is an easier way using the metric. In matrix terminology, the transformation of the metric components is
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where (g) is the matrix of gαβ , (η) of ηαβ , etc., and where ‘T’ denotes transpose. It follows that the determinants satisfy
det (g) = det ( ) det (η) det ( T ). |
(6.12) |
148 |
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Curved manifolds |
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But for any matrix |
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det ( ) = det ( T ), |
(6.13) |
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and we can easily see from Eq. (6.2) that |
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det (η) = −1. |
(6.14) |
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Therefore, we get |
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det (g) = −[det ( )]2. |
(6.15) |
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Now we introduce the notation |
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g := det (gα β ), |
(6.16) |
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which enables us to conclude from Eq. (6.15) that |
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det ( α β ) = (−g)1/2. |
(6.17) |
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Thus, from Eq. (6.9) we get |
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dx0 dx1 dx2 dx3 = [−det (gα β )]1/2dx0 dx1 dx2 |
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= (−g)1/2dx0 dx1 dx2 dx3 . |
(6.18) |
This is a very useful result. It is also conceptually an important result because it is the first example of a kind of argument we will frequently employ, an argument that uses locally flat coordinates to generalize our flat-space concepts to analogous ones in curved space. In this case we began with dx0 dx1 dx2 dx3 = d4x in a locally flat coordinate system. We argue that this volume element at P must be the volume physically measured by rods and clocks, since the space is the same as Minkowski space in this small region. We then find that the value of this expression in arbitrary coordinates {xμ } is Eq. (6.18), (−g)1/2 d4x , which is thus the expression for the true volume in a curved space at any point in any coordinates. We call this the proper volume element.
It should not be surprising that the metric comes into it, of course, since the metric measures lengths. We only need remember that in any coordinates the square root of the negative of the determinant of (gαβ ) is the thing to multiply by d4x to get the true, or proper, volume element.
Perhaps it would be helpful to quote an example from three dimensions. Here proper volume is (g)1/2, since the metric is positive-definite (Eq. (6.14) would have a + sign). In spherical coordinates the line element is dl2 = dr2 + r2dθ 2 + r2 sin2 θ dφ2, so the metric is
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(6.19) |
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Its determinant is r4 sin2 θ , so (g)1/2 d3x is |
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r2 sin θ dr dθ dφ, |
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(6.20) |
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which we know is the correct volume element in these coordinates. |
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