GRAVITATION. AND COSMOLOGY: PRINCIPLES AND APPLICATIONS. OF THE GENERAL THEORY. OF RELATIVITY. STEVEN WEINBERG. Massachusetts. Library of Congress Cataloging in Publication Data: Weinberg, Steven. Gravitation and cosmology Includes bibliographies. 1. General relativity (Physics) . 2. Gravitation And Cosmology: Principles. And Applications Of The General Theory Of Relativity. Steven Weinberg. Page 2. Page 3. Page 4. Page 5. Page 6.

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On the cosmic scale, gravitation dominates the universe. Nuclear and Cosmology, S. Weinberg, Cambridge University Press, Two useful sources of. Gravitation and cosmology principles and applications of the general theory of relativity - Weinberg - Ebook download as PDF File .pdf), Text File .txt) or read book online. OF RELATIVITY STEVEN WEINBERG Massachusetts Institute. astrophysics, gravitation and quantum physics Cheng, Oxford Relativity, Gravitation, and Steven Weinberg -

Newtonian classic gravity admits a geometric description. Together with special relativity, it allows a heuristic description of the general relativity GR.

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The inertial movement in classical mechanics is related to the geometry of space and time, practically along geodesics in which the world lines are straight lines in relativist spacetime. Ehlers Due to the principle of equivalence between inertial and gravitational masses, when considering gravity, no distinction is made between inertial motion and gravity.

This allows the definition of a new class of bodies in a free fall, defining a geometry of space and time by a geodetic motion that depends on the gradient of the gravitational potential. Hence the Newton-Cartan theory, a geometric formula of Newtonian gravity in curved spacetime using only covariant concepts.

Ehlers Havas Newtonian geometric gravity is a limiting case of special relativistic mechanics. Where gravity can be neglected, physics is Lorentzian invariant as in relativity, rather than Galilean invariant as in classical mechanics.

Giulini Lorentz's symmetry involves additional structures through light cones defining a causal structure 1. Together with the world lines for freefalling bodies, light cones can be used to reconstruct the semi-Riemannian spacetime metric, at least up to a positive scalar factor, resulting in a conforming structure or geometry. If gravity is taken into account, the temporal straight lines defining an inertial frame without gravity are curved, resulting in a change in spacetime geometry.

Schutz and Schutz Proper time measured with clocks in a gravitational field does not follow the rules of special relativity it is not measured by the Minkowski metric , requiring a more general, curved geometry of space, with a pseudo-Riemannian metric naturally associated with a certain type of connection, the Levi-Civita connection, which satisfies the principle of equivalence and makes the local space Minkowskian.

Ehlers In November , at the Academy of Sciences of Prussia, Einstein presented the field equations 2 that include gravity, which specifies how space and time geometry is influenced by matter and radiation. GR is a metric theory of gravity.

It is based on Einstein's equations, which describe the relationship between the geometry of a four-dimensional, pseudo-Riemannian manifold, representing spacetime and energy-impulse contained in that spacetime.

Gravity corresponds to changes in space and time properties, which in turn modify the paths of objects. It assumes that the change in gravitational mass o f the apparatus equals the change in its inertial mass and hence its internal energy.

W e conclude then that the metric tensor g must like r]yji have three positive eigenvalues. Newton believed that inertial forces. Einstein considered himself a follower o f Mach.

The fact that g is related to rjap by the congruence 3. The inertial frames. The distinction is not one o f metaphysics but o f physics. It follows that there exists a matrix D.

The celestial bodies play a role here because the gravitational field equations for gfiv need boundary conditions at infinity. Cocconi and Salpeter pointed o u t10 that there is a large mass near us. In this case the three transitions among neighboring states should have the same energy and the photon absorption coefficient should show a single sharp peak at this energy. In this sense. These points are so important that they are worth repeating. Hughes et al.

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity

W e are not yet ready to go into the details o f the field equations and cosmology. See Section In the absence o f nearby matter. This was checked experimentally by Hughes. I f we think o f the L i7 nucleus as a single proton with angular momentum.

The issue between Mach and Einstein can be drawn by asking whether in fact the presence o f large nearby masses does affect the laws o f motion. When a large mass like the sun is brought close. The Theory of Relativity Clarendon Press. Chapter VII. Wapstra and G. The General Theory Interscience Publishers. Kemmer 2nd rev. Englewood Cliffs. Brose Dover Publications.

Wiese and D. Blamont and F.

L59 W itten Wiley. On the experimental tests o f the Principle o f Equivalence. Introduction to the Theory of Relativity Prentice-Hall. Pound and J.

Space Sei. Pound and G. The Theory of Space. Greenstein and V. New York.. For other white dwarfs. For a review and earlier references. Spectroscopic studies o f Sirius B have very recently yielded an estimate 2.

Turnbull and A. Cocconi and E. W e could continue to follow this approach. W e wrote down the equations that hold. I f Bread or Butter wanted weight. Resolve by Signes and Tangents straight. Than Tycho Brahe or Erra Pater: For he by Geometrick scale Could take the size of Pots of Ale. This chapter is devoted to an outline o f the language common to both. Sir Hudibras. This method is based on an alternative version o f the Principle o f Equivalence. The meaning o f general covariance can be brought forward b y comparing it with Lorentz invariance.

But at any given point there is a class o f coordinate systems. The difference is that we do not require that these quantities drop out at the end. To put this briefly: The Principle o f General Covariance is not an invariance principle. The equation is generally covariant. It should be stressed that general covariance by itself is empty o f physical content.

Condition 1 then tells us that our equation holds in these systems. The requirement that this velocity not appear in the transformed equation is what we call the Principle o f Special Relativity. The equation holds in the absence o f gravitation. The significance o f the Principle o f General Covariance lies in its statement about the effects o f gravitation. Just as any equation can be made generally covariant. W e can. For other quantities. W ith this understanding we shall see in this and the next chapter that the Principle o f General Covariance makes an unambiguous statement about the effects of gravitational fields on any system.

For some quantities. W e shall return to the analogy between general relativity and electrodynamics several times in following chapters. In this section we describe one class of objects whose transformation properties are particularly simple. The Principle o f General Covariance can only be applied on a scale that is small compared with the space-time distances typical o f the gravitational field.

There are in general many generally covariant equations that reduce to a given special-relativistic equation in the absence o f gravitation. The next simplesttransformation rule is that o f acontravariant vector. A tensor with upper indices jti. The obvious example is a pure number. A very closely related transformation rule is that o f a covariant vector U.

Another example is the proper time dr. From the contravariant and covariant vectors we can immediately generalize to the tensors. W e can now recognize one very large class of invariant equations: Any equation will be invariant under general coordinate transformations if it states the equality o f two tensors with the same upper and lower indices.

A vector is just a tensor with one index and a scalar is a tensor with no indices. In a different coordinate system x. Its inverse is a contravariant tensor.

This is accomplished through a few simple algebraic operations: A Linear Combinations. The product of the components o f two tensors yields a tensor whose upper and lower indices consist o f all the upper and lower indices o f the two original tensors.

B p then p is a tensor.

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In contrast. Setting an upper and lower index equal and summing it over its four values yields a new tensor with these two indices absent. Tp dxp d x. Note that lowering an index and then raising it again gives back the original tensor. Since these are all physically equivalent. For the sake o f completeness. I f we take the direct product o f a contravariant or mixed tensor T with the metric tensor g and contract the index fi with one of the contravariant indices of T.

One particularly important combined operation results in the raising and lowering of indices. I have not yet mentioned differentiation.

The reader will have noticed that this discussion o f tensor algebra is precisely the same as the corresponding discussion in the chapter on special relativity see Section 2.

In Section 6 Ave shall see that there is a kind o f differentiation. This is because the derivative o f a tensor is in general not a tensor. A quantity such as g. To determine the proportionality constant. To define this quantity in a general coordinate system.

There is one tensor density whose components are the same in all. W e can also form a covariant density by lowering indices in the usual way.

The rules o f tensor algebra may be easily extended to encompass tensor densities. A The linear combination o f two tensor densities o f the same weight W is a tensor density o f weight W.

We recall its definition. From B and C it follows that raising and lowering indices does not change the weight of a tensor density. C The contraction o f indices on a tensor density o f weight W yields a tensor density of weight W.

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Passing from x Mto a different system x. In this system there can be no gravitational force on free particles. Differentiate the identity. Note that dxp dxa A dx.

The Principle of General Covariance then tells us that 4. Using Eq. W e recall its transformation rule y Differentiate with respect to x 'v: It is obvious how these definitions are to be extended to a general tensor. W e can also extend the idea o f co variant differentiation to tensor densities.

Its co variant derivative is also a tensor. The effect is that the covariant derivative with respect to xp o f a tensor density J o f weight W is constructed just as if it were an ordinary tensor. In particular: A The covariant derivative o f a linear combination of tensors with constant coefficients is the same linear combination of the covariant derivatives. We can also show in the same way that thecovariant derivatives o f the other forms o f the metric tensor also vanish.

We also note that the covariant derivative o f the metric tensor is zero. Theseproperties suggest the following algorithm for assessing the effects of gravitation on physical systems: It converts tensors to other tensors. Simplest of all is. W e can also use 4. In Det M. To prove 4. Applying 4. These are three-dimensional coordinate systems characterized by the condition that gt. Consider first a contra variant vector A'l r. Obvious examples come to mind. Any tensor can be defined along a curve by parallel transport by requiring its covariant derivative along the curve to vanish.

It should be mentioned that the covariant derivative of a tensor field along a curve may be determined from its ordinary covariant derivative. The vector A is then subject to the first-order differential equations d.

Also note that the electromagnetic potential A x obeys an inhomogeneous gauge transformation law. As such. How are we to construct gauge-invariant equations? We see in Chapter 7 that the field equations for gravitation are constructed in an analogous manner.

That this is gauge-invariant is obvious. A proper explanation o f this point would fill another book. The analogy between the gauge invariance o f electrodynamics and the general covariance o f general relativity can be extended to a similar dynamic symmetry. In order to deal with these properties in a unified way. II even 4. A covariant tensor o f rank p. In n dimensions. This section presents some o f the fundamental results o f the theory o f differential forms.

As already indicated. The partial derivative operator djdx11 is a covariant vector. The associativity and commutativity rules 4. Among the special cases o f this lemma are two well-known results o f three-dimensional vector analysis. From the associativity and commutativity rules 4.

The easiest way to see this is just to note that the partial derivatives used to define the exterior derivative can be replaced with covariant derivatives. Our previously derived results 4. The answer is yes. Just as the exterior derivative provides a natural generalization o f the familiar gradient. W ith this understanding. For simplicity o f notation. The simplest example of an integral o f the form 4.

W e shall actually be concerned here with what are called orientable manifolds. Taking account o f the antisymmetry of t. Thus the whole integral either is unchanged or changes by a minus sign. This result shows. For this approach to chiral symmetry. W ith this general definition o f the integrals of jo-forms. In the next simplest case. Symmetries and Reflections Indiana University Press. See Section 4.

Sections 1. Electromagnetic Theory McGraw-Hill. Differential Forms Academic Press. For an extremely readable text on the theory o f differential forms. Alexander Pope. Or dip their pinions in the painted bow Or blow fierce tempests on the wintry main.

Some less refined. Or suck the mists in grosser air below. The resulting equations will be generally covariant and true in the absence o f gravitation. Or roll the planets through the boundless sky. The technique to be used is that afforded by the Principle of General Covariance: We must first write the equations as they hold in special relativity. Equation 5. Since the familiar equation for free fall. W e recognize in Eqs.

For the present. In this event. W e can then make 5. It is instructive to evaluate the current vector J v. In special relativity it is 5. The electromagnetic force on a particle o f charge e is given in the absence o f gravitation by Eq. In some works it is this scalar that is defined as the delta function.

Following precisely the samereasoning as we did for J a in the last section. For a system o f point particles the special-relativistic energy-momentum tensor is given in Section 2. Just as we would expect. Then the generally covariant equation that agrees with 5. For an isolated system. Note that p and p are always defined as the pressure and energy density measured by an. Returning for a moment to the purely material energy-momentum tensor 5. Since it is not moving. This equation is soluble if p is given as a function o f p.

V2p 5. As in the last chapter. Thus we must address ourselves to the question: What tensors can be formed from the metric tensor and its derivatives? In this chapter we treat this as a purely mathematical problem.. If we use only g and its first derivatives. To do this.. Differentiation with respect to xK gives d3x.

O f course. Suppose that our particle has a characteristic linear dimension d.

The answer again is one o f scale. The existence o f the tensor R xllVK raises once again the question o f whether or not the Principle o f Equivalence or the Principle o f General Covariance uniquely determines the effects o f gravitation on arbitrary physical systems. The Riemann-Christoffel tensor has one more derivative o f the metric than the affine connection.

H ow then can we tell whether 6. But this is an equality between tensors. Any quantity that transforms as a tensor under general coordinate transformations will have to transform as a tensor under this limited class o f transformations. W e see from Eq. Since the affine connection vanishes at X. The change in 8 when parallel-transported around C can be written as the sum o f the changes in S when transported around each o f these little curves.

I f the curve is small enough. The most prominent are the contracted forms. Sav Sbp However. Consider a closed curve consisting of two segments A and B joining points x 11 and X M. It follows that. The change in a vector when parallel-transported from x to X along A must be canceled by the change in S when parallel-transported along B from X to x.

How can we tell if the space is really permeated by a gravitational field. The answer is contained in the following theorem: In this approach Eq. In other words. The necessity o f these two conditions is obvious.

In this coordinate system the metric. To prove the sufficiency o f Eq. But the vanishing o f a tensor is an invariant statement. Then to satisfy 6. But 6. Consider the second covariant derivative o f a covariant vector Vx: A Symmetry: C eliminates the one other scalar that might have been formed in four dimensions. From 6. To count the number o f algebraically independent components o f R XflVK. It may strike the reader as odd that a curved line should have zero curvature.

Contracting X with v gives the Ricci tensor 6. Equation 1. Since 6. This is also the number o f independent components o f the Ricci tensor R in three dimensions. In three dimensions 6. B y using the covariance. This exception does not occur for higher dimensional spaces.

The latter name is used because the necessary and sufficient condition for the existence o f a coordinate system in which g is proportional to a constant matrix throughout.

Petrov2 has given an equivalent description o f the four nonvanishing curvature invariants as roots o f a secular equation, and has classified various algebraic types o f W eyl tensor according to the degeneracies o f these roots. Finally, it should be emphasized that 6. There are in general differential relations among these invariants, and the number o f functionally independent curvature invariants is less than 6.

The curvature tensor obeys important differential identities, in addition to the algebraic identities discussed in Section 6. Then at x, Eq. By permuting v, we obtain the Bianchi identities.

These equations are manifestly generally covariant, so since they hold in locally inertial systems they hold in general. They can also, o f course, be checked by direct calculation. W e shall be particularly concerned with the contracted form o f 6. W e have seen in this chapter that the nonvanishing o f the tensor R XflVK is the true expression o f the presence o f a gravitational field. It is therefore not surprising that Einstein and his successors have regarded the effects o f a gravitational field as producing a change in the geometry o f space and time.

The reader should be warned that these views are heterodox and would meet with objections from many general relativists. From Eq. The introduction o f the curvature tensor was motivated here by the need to construct suitable field equations for the gravitational field.

However, the curvature tensor is also useful in expressing the effects o f gravitation on physical systems. The equations o f motion are. Although a freely falling particle appears to be at rest in a coordinate frame falling with the particle, a pair o f nearby freely falling particles will exhibit a relative motion that can reveal the presence o f a gravitational field to an observer that falls with them.

This is o f course not a violation o f thePrincipleo f Equivalence, because the effect o f the right-hand side o f 6. Also, see Bibliography, Chapter 3. See, for example, L. Petrov, Uch. Kazan Gos. Kelleher Pergamon Press, Oxford, , Chapter 3. W eyl, Mat. Reference 4, Section Therefore I am not going to defend it with a single word.

Sommerfeld, February 8, Chapters 3 through 5 have provided us with one-half o f a complete theory o f gravitation, that is, with a mathematical description o f gravitational fields that dictates their effects on arbitrary physical systems. In this chapter we move on to the second half o f general relativity, that is, to the differential equations that determine the gravitational fields themselves. The field equations for gravitation are inevitably going to be more complicated than those for electromagnetism.

That is, the gravitational field equations will have to be nonlinear partial differential equations, the nonlinearity representing the effect o f gravitation on itself. In dealing with these nonlinear effects we are guided once again by the Principle o f Equivalence. At any point X in an arbitrarily strong gravitational field, we can define a locally inertial coordinate system such that 7.

Hence for x near X , the metric tensor gap can differ from rjxp only by terms quadratic in x — X. In this coordinate system the gravitational field is weak near X , and we can hope to describe the field by linear partial differential equations.

And once we know what these weak-field equations are, we can find the general field equations by reversing the coordinate transformation that made the field weak. Unfortunately, we have very little empirical information about the weakfield equations. This is not for any fundamental reason, but rather because gravitational radiation is so weakly generated and absorbed by matter, that it has not yet certainly been detected. However, although forgivable, our ignorance does prevent us from proceeding as directly as we did in previous chapters, and some guesswork will be unavoidable.

First let us recall that in a weak static field produced by a nonrelativistic mass density p, the time-time component o f the metric tensor is approximately given by 9oo -. This field equation is only supposed to hold for weak static fields generated by nonrelativistic matter, and is not even Lorentz invariant as it stands. However, 7. Let us imagine G to be expanded in a sum o f products o f derivatives o f the metric, and classify each term according to the total number N o f derivatives o f.

Let us review what we know about the left-hand-side o f the field equation 7. E For a weak stationary field produced by nonrelativistic matter the 00 component o f 7. W e saw in Section 6. This is automatically symmetric [see Eq. Using the Bianchi identity 6. W e can reject the second possibility, because 7. Thus if R. Finally, we use theproperty E to fix the constant C system always has IT,,!

Tnn , so we are concerned here. When the field is static all time derivatives vanish, and the components we need become 7? Hence 7. O f course we can also go from 7. In a vacuum T vanishes, so from 7. See Section 6. It is only in four or more dimensions that true gravitational fields can exist in empty space.

The freedom to use first derivatives does not allow any new terms in 0 see Section 6. Reference to the precise definitions 3. Such a tensor could be constructed by writing 7. To determine a l and bx we pass to the Newtonian limit.

For a static field, 7. Repeated Latin indices are summed over the values 1, 2, 3. Such theories have been. See Chapter The first term o f Eq.

I f co is much larger than unity. It must be stressed that the role o f the scalar field in the Brans-Dicke theory is confined to its effect on the gravitational field equations.

Throughout most o f this book it will be assumed that there is no scalar field cf that contributes to long-range interactions. Although algebraically independent. When written in terms o f the vector potential. This degree of freedom o f course corresponds to gauge invariance. The unknown metric tensor also has 10 algebraically independent components.

Although we cannot write them as covariant equations. The condition 7. The four conditions 7. The choice o f a coordinate system can be expressed in four coordinate conditions. One particularly convenient choice o f a coordinate system is represented by the harmonic coordinate conditions 3 g 'T i. In the absence o f gravitational fields. A t first sight this looks feasible.

Weinberg defines and proves in his review on the Cosmological Constant The gravitational coupling strength is given by. Gravitation and Cosmology : Principles and Applications of the Which are some good books on Theory of Relativity? Go on modifying the pdf filename to Ch02Rel. Renormalization group and the Planck scale - RoyalSocietyPublishing It is well Steven Weinberg's seminal proposal, more than 30 years ago, that a quantum theory of Gravitational waves in cold dark matter ; Jun 7, Raphael Flauger and Steven Weinberg.

The Value of Einstein's Mistakes ; Apr 12, Steven Weinberg's love Steven Weinberg analyzes some of. Gravity - Wikipedia ; Gravity from Latin gravitas, meaning 'weight' , or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light —are brought toward or gravitate toward one another. On Earth, gravity gives weight to physical objects, and the Moon's gravity causes the ocean tides.

The gravitational attraction of the original gaseous matter Inflation cosmology - Wikipedia ; In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. Following the inflationary period, the universe continues to expand, but at Hartle Addison Wesley, Cosmology - ustc. Principles And Weinberg, Gravitation and Cosmology and readers.It follows then from the calculation o f P x in E above that the total energy and momentum are equal to the sum o f the values P x for each subsystem alone.

It must be stressed that the role o f the scalar field in the Brans-Dicke theory is confined to its effect on the gravitational field equations. Einstein, Phys. Related Papers. Letters, 19, Attendance to the lectures is not compulsory, but if you come I ask you to pay attention and not disrupt the class with personal conversation. Ehlers Havas Newtonian geometric gravity is a limiting case of special relativistic mechanics. Write a customer review. site Second Chance Pass it on, trade it in, give it a second life.

For the present.

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