(E-Books) Stephen Hawking - The Nature Of Space And Time

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hep-th/9409195 30 Sep 94

1. Classical Theory

S. W. Hawking

In these lectures Roger Penrose and I will put forward our related but rather diﬀerent

viewpoints on the nature of space and time. We shall speak alternately and shall give three

lectures each, followed by a discussion on our diﬀerent approaches. I should emphasize that

these will be technical lectures. We shall assume a basic knowledge of general relativity

and quantum theory.

There is a short article by Richard Feynman describing his experiences at a conference

on general relativity. I think it was the Warsaw conference in 1962. It commented very

unfavorably on the general competence of the people there and the relevance of what

they were doing. That general relativity soon acquired a much better reputation, and

more interest, is in a considerable measure because of Roger’s work. Up to then, general

relativity had been formulated as a messy set of partial diﬀerential equations in a single

coordinate system. People were so pleased when they found a solution that they didn’t

care that it probably had no physical signiﬁcance. However, Roger brought in modern

concepts like spinors and global methods. He was the ﬁrst to show that one could discover

general properties without solving the equations exactly. It was his ﬁrst singularity theorem

that introduced me to the study of causal structure and inspired my classical work on

singularities and black holes.

I think Roger and I pretty much agree on the classical work. However, we diﬀer in

our approach to quantum gravity and indeed to quantum theory itself. Although I’m

regarded as a dangerous radical by particle physicists for proposing that there may be loss

of quantum coherence I’m deﬁnitely a conservative compared to Roger. I take the positivist

viewpoint that a physical theory is just a mathematical model and that it is meaningless

to ask whether it corresponds to reality. All that one can ask is that its predictions should

be in agreement with observation. I think Roger is a Platonist at heart but he must answer

for himself.

Although there have been suggestions that spacetime may have a discrete structure

I see no reason to abandon the continuum theories that have been so successful. General

relativity is a beautiful theory that agrees with every observation that has been made. It

may require modiﬁcations on the Planck scale but I don’t think that will aﬀect many of

the predictions that can be obtained from it. It may be only a low energy approximation

to some more fundemental theory, like string theory, but I think string theory has been

over sold. First of all, it is not clear that general relativity, when combined with various

other ﬁelds in a supergravity theory, can not give a sensible quantum theory. Reports of

the death of supergravity are exaggerations. One year everyone believed that supergravity

was ﬁnite. The next year the fashion changed and everyone said that supergravity was

bound to have divergences even though none had actually been found. My second reason

for not discussing string theory is that it has not made any testable predictions. By

contrast, the straight forward application of quantum theory to general relativity, which I

will be talking about, has already made two testable predictions. One of these predictions,

the development of small perturbations during inﬂation, seems to be conﬁrmed by recent

observations of ﬂuctuations in the microwave background. The other prediction, that

black holes should radiate thermally, is testable in principle. All we have to do is ﬁnd a

primordial black hole. Unfortunately, there don’t seem many around in this neck of the

woods. If there had been we would know how to quantize gravity.

Neither of these predictions will be changed even if string theory is the ultimate

theory of nature. But string theory, at least at its current state of development, is quite

incapable of making these predictions except by appealing to general relativity as the low

energy eﬀective theory. I suspect this may always be the case and that there may not be

any observable predictions of string theory that can not also be predicted from general

relativity or supergravity. If this is true it raises the question of whether string theory is a

genuine scientiﬁc theory. Is mathematical beauty and completeness enough in the absence

of distinctive observationally tested predictions. Not that string theory in its present form

is either beautiful or complete.

For these reasons, I shall talk about general relativity in these lectures. I shall con-

centrate on two areas where gravity seems to lead to features that are completely diﬀerent

from other ﬁeld theories. The ﬁrst is the idea that gravity should cause spacetime to have

a begining and maybe an end. The second is the discovery that there seems to be intrinsic

gravitational entropy that is not the result of coarse graining. Some people have claimed

that these predictions are just artifacts of the semi classical approximation. They say that

string theory, the true quantum theory of gravity, will smear out the singularities and will

introduce correlations in the radiation from black holes so that it is only approximately

thermal in the coarse grained sense. It would be rather boring if this were the case. Grav-

ity would be just like any other ﬁeld. But I believe it is distinctively diﬀerent, because

it shapes the arena in which it acts, unlike other ﬁelds which act in a ﬁxed spacetime

background. It is this that leads to the possibility of time having a begining. It also leads

to regions of the universe which one can’t observe, which in turn gives rise to the concept

of gravitational entropy as a measure of what we can’t know.

In this lecture I shall review the work in classical general relativity that leads to these

ideas. In the second and third lectures I shall show how they are changed and extended

when one goes to quantum theory. Lecture two will be about black holes and lecture three

will be on quantum cosmology.

The crucial technique for investigating singularities and black holes that was intro-

duced by Roger, and which I helped develop, was the study of the global causal structure

of spacetime.

Time

Space

Null geodesics through p

generating part of

Null geodesic in (p) which

does not go back to p and has

no past end point

Point removed

from spacetime

Chronological

future

I(p)

I(p) +

Deﬁne I+(p) to be the set of all points of the spacetime Mthat can be reached from pby

future directed time like curves. One can think of I+(p) as the set of all events that can

be inﬂuenced by what happens at p. There are similar deﬁnitions in which plus is replaced

by minus and future by past. I shall regard such deﬁnitions as self evident.

I(S)

timelike curve

I(S)

I(S) can't be timelike

I(S)

I(S) can't be spacelike

All timelike curves from q leave +

I(S)

One now considers the boundary ˙

I+(S)ofthefutureofasetS. It is fairly easy to

see that this boundary can not be time like. For in that case, a point qjust outside the

boundary would be to the future of a point pjust inside. Nor can the boundary of the

future be space like, except at the set Sitself. For in that case every past directed curve

from a point q, just to the future of the boundary, would cross the boundary and leave the

future of S. That would be a contradiction with the fact that qis in the future of S.

I(S)

null geodesic segment in

I(S)

null geodesic segment in

I(S)

future end point of generators of

One therefore concludes that the boundary of the future is null apart from at Sitself.

More precisely, if qis in the boundary of the future but is not in the closure of Sthere

is a past directed null geodesic segment through qlying in the boundary. There may be

more than one null geodesic segment through qlying in the boundary, but in that case q

will be a future end point of the segments. In other words, the boundary of the future of

Sis generated by null geodesics that have a future end point in the boundary and pass

into the interior of the future if they intersect another generator. On the other hand, the

null geodesic generators can have past end points only on S. It is possible, however, to

have spacetimes in which there are generators of the boundary of the future of a set Sthat

never intersect S. Such generators can have no past end point.

A simple example of this is Minkowski space with a horizontal line segment removed.

If the set Slies to the past of the horizontal line, the line will cast a shadow and there

will be points just to the future of the line that are not in the future of S. There will be

a generator of the boundary of the future of Sthat goes back to the end of the horizontal

I(S)

line removed from

Minkowski space

generator of (S)

with no end point on S

generators of (S)

with past end point on S

line. However, as the end point of the horizontal line has been removed from spacetime,

this generator of the boundary will have no past end point. This spacetime is incomplete,

but one can cure this by multiplying the metric by a suitable conformal factor near the

end of the horizontal line. Although spaces like this are very artiﬁcial they are important

in showing how careful you have to be in the study of causal structure. In fact Roger

Penrose, who was one of my PhD examiners, pointed out that a space like that I have just

described was a counter example to some of the claims I made in my thesis.

To show that each generator of the boundary of the future has a past end point on

the set one has to impose some global condition on the causal structure. The strongest

and physically most important condition is that of global hyperbolicity.

∩

I(p) _

I(q)

An open set Uis said to be globally hyperbolic if:

1) for every pair of points pand qin Uthe intersection of the future of pand the past

of qhas compact closure. In other words, it is a bounded diamond shaped region.

2) strong causality holds on U. That is there are no closed or almost closed time like

curves contained in U.

every timelike curve

intersects (t)

(t)

The physical signiﬁcance of global hyperbolicity comes from the fact that it implies

that there is a family of Cauchy surfaces Σ(t)forU. A Cauchy surface for Uis a space

like or null surface that intersects every time like curve in Uonce and once only. One can

predict what will happen in Ufrom data on the Cauchy surface, and one can formulate a

well behaved quantum ﬁeld theory on a globally hyperbolic background. Whether one can

formulate a sensible quantum ﬁeld theory on a non globally hyperbolic background is less

clear. So global hyperbolicity may be a physical necessity. But my view point is that one

shouldn’t assume it because that may be ruling out something that gravity is trying to

tell us. Rather one should deduce that certain regions of spacetime are globally hyperbolic

from other physically reasonable assumptions.

The signiﬁcance of global hyperbolicity for singularity theorems stems from the fol-

lowing.

geodesic of

maximum length

Let Ube globally hyperbolic and let pand qbe points of Uthat can be joined by a

time like or null curve. Then there is a time like or null geodesic between pand qwhich

maximizes the length of time like or null curves from pto q. The method of proof is to

show the space of all time like or null curves from pto qis compact in a certain topology.

One then shows that the length of the curve is an upper semi continuous function on this

space. It must therefore attain its maximum and the curve of maximum length will be a

geodesic because otherwise a small variation will give a longer curve.

geodesic

point conjugate

to p along neighbouring

geodesic

non-minimal

geodesic minimal geodesic

without conjugate points

point conjugate to p

One can now consider the second variation of the length of a geodesic γ. One can show

that γcan be varied to a longer curve if there is an inﬁnitesimally neighbouring geodesic

from pwhich intersects γagain at a point rbetween pand q.Thepointris said to be

conjugate to p. One can illustrate this by considering two points pand qon the surface of

the Earth. Without loss of generality one can take pto be at the north pole. Because the

Earth has a positive deﬁnite metric rather than a Lorentzian one, there is a geodesic of

minimal length, rather than a geodesic of maximum length. This minimal geodesic will be

a line of longtitude running from the north pole to the point q. But there will be another

geodesic from pto qwhich runs down the back from the north pole to the south pole and

then up to q. This geodesic contains a point conjugate to pat the south pole where all the

geodesics from pintersect. Both geodesics from pto qare stationary points of the length

under a small variation. But now in a positive deﬁnite metric the second variation of a

geodesic containing a conjugate point can give a shorter curve from pto q.Thus,inthe

example of the Earth, we can deduce that the geodesic that goes down to the south pole

and then comes up is not the shortest curve from pto q. This example is very obvious.

However, in the case of spacetime one can show that under certain assumptions there

ought to be a globally hyperbolic region in which there ought to be conjugate points on

every geodesic between two points. This establishes a contradiction which shows that the

assumption of geodesic completeness, which can be taken as a deﬁnition of a non singular

spacetime, is false.

The reason one gets conjugate points in spacetime is that gravity is an attractive force.

It therefore curves spacetime in such a way that neighbouring geodesics are bent towards

each other rather than away. One can see this from the Raychaudhuri or Newman-Penrose

equation, which I will write in a uniﬁed form.

Raychaudhuri - Newman - Penrose equation

dρ

dv =ρ2+σij σij +1

nRablalb

where n= 2 for null geodesics

n= 3 for timelike geodesics

Here vis an aﬃne parameter along a congruence of geodesics, with tangent vector la

which are hypersurface orthogonal. The quantity ρis the average rate of convergence of

the geodesics, while σmeasures the shear. The term Rablalbgives the direct gravitational

eﬀect of the matter on the convergence of the geodesics.

Einstein equation

Rab −1

2gabR=8πTab

Weak Energy Condition

Tabvavb≥0

for any timelike vector va.

By the Einstein equations, it will be non negative for any null vector laif the matter obeys

the so called weak energy condition. This says that the energy density T00 is non negative

in any frame. The weak energy condition is obeyed by the classical energy momentum

tensor of any reasonable matter, such as a scalar or electro magnetic ﬁeld or a ﬂuid with

a reasonable equation of state. It may not however be satisﬁed locally by the quantum

mechanical expectation value of the energy momentum tensor. This will be relevant in my

second and third lectures.

Suppose the weak energy condition holds, and that the null geodesics from a point p

begin to converge again and that ρhas the positive value ρ0. Then the Newman Penrose

equation would imply that the convergence ρwould become inﬁnite at a point qwithin an

aﬃne parameter distance 1

ρ0if the null geodesic can be extended that far.

If ρ=ρ0at v=v0then ρ≥1

ρ−1+v0−v. Thus there is a conjugate point

before v=v0+ρ−1.

pneighbouring geodesics

meeting at q

future end point

of in (p)

crossing region

of light cone

inside (p)

γ+

Inﬁnitesimally neighbouring null geodesics from pwill intersect at q. This means the point

qwill be conjugate to palong the null geodesic γjoining them. For points on γbeyond

the conjugate point qthere will be a variation of γthat gives a time like curve from p.

Thus γcan not lie in the boundary of the future of pbeyond the conjugate point q.Soγ

will have a future end point as a generator of the boundary of the future of p.

The situation with time like geodesics is similar, except that the strong energy con-

dition that is required to make Rablalbnon negative for every time like vector lais, as

its name suggests, rather stronger. It is still however physically reasonable, at least in an

averaged sense, in classical theory. If the strong energy condition holds, and the time like

geodesics from pbegin converging again, then there will be a point qconjugate to p.

Finally there is the generic energy condition. This says that ﬁrst the strong energy

condition holds. Second, every time like or null geodesic encounters some point where

Strong Energy Condition

Tabvavb≥1

2vavaT

there is some curvature that is not specially aligned with the geodesic. The generic energy

condition is not satisﬁed by a number of known exact solutions. But these are rather

special. One would expect it to be satisﬁed by a solution that was ”generic” in an appro-

priate sense. If the generic energy condition holds, each geodesic will encounter a region

of gravitational focussing. This will imply that there are pairs of conjugate points if one

can extend the geodesic far enough in each direction.

The Generic Energy Condition

1. The strong energy condition holds.

2. Every timelike or null geodesic contains a point where l[aRb]cd[elf]lcld6=0.

One normally thinks of a spacetime singularity as a region in which the curvature

becomes unboundedly large. However, the trouble with that as a deﬁnition is that one

could simply leave out the singular points and say that the remaining manifold was the

whole of spacetime. It is therefore better to deﬁne spacetime as the maximal manifold on

which the metric is suitably smooth. One can then recognize the occurrence of singularities

by the existence of incomplete geodesics that can not be extended to inﬁnite values of the

aﬃne parameter.

Deﬁnition of Singularity

A spacetime is singular if it is timelike or null geodesically incomplete, but

can not be embedded in a larger spacetime.

This deﬁnition reﬂects the most objectionable feature of singularities, that there can be

particles whose history has a begining or end at a ﬁnite time. There are examples in which

geodesic incompleteness can occur with the curvature remaining bounded, but it is thought

that generically the curvature will diverge along incomplete geodesics. This is important if

one is to appeal to quantum eﬀects to solve the problems raised by singularities in classical

general relativity.

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hep-th/940919530Sep941.ClassicalTheoryS.W.HawkingIntheselecturesRogerPenroseandIwillputforwardourrelatedbutratherdierentviewpointsonthenatureofspaceandtime.Weshallspeakalternatelyandshallgivethreelectureseach,followedbyadiscussiononourdierentapproaches.Ishouldemphasizethatthesewillbetechnicallectu...

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