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30 Scientific American, November 2010
Photograph/Illustration by Artist Name
T. Padmanabhan is a theoretical physicist at IUCAA, Pune. His
work on gravity has won international acclaim, including the First
Award from the Gravity Research Foundation, USA. He was a
Sackler Distinguished Astronomer at Cambridge and is currently
the President of the Cosmology Commission of the IAU.
Grappling With
Gravity
p h ys i c s
The underlying description of gravity may lie in a microstructure made
up of “atoms of spacetime”
By Thanu Padmanabhan
T
he first hint that gravity is an odd-man-out
among the fundamental forces came when Einstein realized that the ideas of Newtonian gravity need to be modified to make them compatible with the special theory of relativity. The special theory of relativity demands that no
influence can propagate with a speed greater
than that of light. So, if the Sun disappears at some instant, its
gravitational influence on Earth can disappear only after a delay
of at least eight minutes because light takes about eight minutes to
reach the Earth from the Sun. On the other hand, according to
Newton’s theory of gravity, the gravitational force will disappear
instantaneously. The attempt to correct such inconsistencies be-
tween Newtonian gravity and special relativity led Einstein to the
general theory of relativity which interprets gravity as a manifestation of the curvature of spacetime. In this approach, verified by
numerous experiments, gravity is described in terms of what is
called the spacetime metric, which is a mathematical object that
essentially allows one to compute all the geometrical properties of
the spacetime, as well as the influence of gravity on other physical
systems. Einstein also introduced a set of equations which allow
us to determine this metric from the distribution of matter and
energy which acts as the source for gravity.
Every massive body curves the spacetime around it and other
bodies move in this curved geometry along ‘straight lines’ which
are defined to be the shortest paths between two events. The no-
in brief
The outstanding problem in physics is to
combine Einstein’s description of gravity
with quantum theory. Such a theory is
needed for the mathematical consistency
of theoretical physics.
Gravity is today understood as the manifestation of the curvature of spacetime.
So, it is not really a force, unlike electricity
or magnetism.
A conceptual rather than technical
breakthrough may be needed. Instead of
the conventional view that gravity is a
fundamental interaction, it could be that a
microstructure—the “atoms of space
time”—give rise to it.
The idea chimes with thermodynamics,
where if you can heat something, it has a
microstructure. Recent work suggests
this may be more than an analogy—it
could be a physical reality.
Photo illustration by Rajat Baran
w w w. s c i a m . c o . i n S C I E N T I F I C A M E R I C A N I n d i a 31
T h e D av i e s - U n r u h E f f e c t
Turning Up the Heat
Paul Davies and Bill Unruh independently discovered, in the mid-seventies, a curious result which arises when you combine the principles of general relativity and
quantum theory. They showed that, if you run with an acceleration carrying a
thermometer, it will show a reading different from the one when you are at rest.
This result plays a crucial role in the emergent perspective of gravity.
One usually thinks of the temperature of a system as an absolute entity,
independent of who is measuring it. Further, the temperature of the vacuum,
one would have thought, is zero for all observers. Not so.
The vacuum will appear as a zero temperature state for an observer at
rest; but one who is accelerating through the vacuum will attribute to it a nonzero temperature. Why does this happen? Vacuum, in the language of quantum theory, is not just empty space. It contains a bristling crowd of virtual
particles characterizing the vacuum fluctuations of the quantum field. For
example, the vacuum state harbors virtual photons, which are the quanta of
electromagnetic radiation. These are directly responsible for some of the subtlest effects in quantum electrodynamics measured in the laboratory. But the
pattern of vacuum fluctuations is such that a detector for photons will not be
excited by these fluctuations as long as the detector is at rest or when it is moving with a uniform velocity.
But, as Unruh found, it is a different story if the detector is accelerating.
The vacuum exhibits a different pattern of fluctuations as seen by the accelerated detector and the detector will now get excited. In fact, the work of Davies
and Unruh shows that an accelerated detector will see a spectrum of photons
which is identical to what an unaccelerated detector will detect when
immersed in a hot oven at some temperature. Such an oven—and in fact, every
hot object at some temperature—emits thermal radiation of a specific pattern
which depends only on the temperature. The pattern of vacuum fluctuations as
seen by the accelerated observer mimics this radiation. The temperature perceived by the detector is proportional to the acceleration. Unfortunately, it is
very tiny; an acceleration of 1g produces a temperature of about 10−65 Kelvin,
which is immeasurably small. But, in principle, a person accelerating through
the vacuum carrying a glass of water will find that the water heats up.
tion of a straight line on a curved surface (say, on the surface of
sphere) is quite different from our intuitive notion of a straight
line based on the ideas of flat surfaces. Such a motion appears like
the one produced by a force between the two bodies which we call
the gravitational force. Thus, as we understand it today, gravity is
the manifestation of the curvature of spacetime. In this sense,
gravity clearly stands apart and has a geometric interpretation
unlike, say, electricity or magnetism.
Einstein’s ideas work fine as long as you are in the domain of
classical physics. But the exact laws of physics appear to be quantum mechanical rather than classical. At microscopic scales when
we are dealing with molecules, atoms, nuclei, elementary particles or when studying physics at sufficiently high energies—like in
the earliest moments in the evolution of the universe—we need to
use the laws of quantum theory. When one incorporates the principles of quantum theory into the known classical laws of, say,
electricity and magnetism, we end up getting a more fundamental
description known as quantum electrodynamics. This theory—
verified by experiments to enormous accuracy—is a successful
prototype showing how to combine the principles of quantum
theory with a known classical law for force.
Similarly, we need to combine Einstein’s description of gravity,
given in terms of a curved spacetime, with the principles of quantum theory, thereby obtaining a quantum theory of gravity. One
can, in fact, show that a situation in which gravity is described by
classical laws while the matter is described by quantum laws will
be mathematically inconsistent. Hence quantum gravity is needed for the mathematical consistency of theoretical physics.
The trouble is that a significant amount of effort during the
past half century has not resulted in any conclusive model for
quantum gravity.
A NEW PARADIGM
recent research suggests an intriguing possibility. It appears
that we need yet another fundamental revision in our point of
view regarding gravity: Gravity could just be an emergent phe-
32 S C I E N T I F I C A M E R I C A N I n d i a
—T.P..
nomenon. What is an emergent phenomenon? Simple examples of emergent phenomena in physics include elasticity or
gas dynamics. The laws of elasticity or gas dynamics, expressed
as mathematical equations, were known for centuries. They
are quite adequate for understanding, say, the elastic vibrations of a steel wire used in a suspension bridge or the flow of
air past an airplane wing—without ever referring to the individual atoms which make up the steel wire or air. We say that
gas dynamics or elasticity is an emergent phenomenon while
the really fundamental description is in terms of the atoms
making up the matter.
This has implications when you move from the classical to
the quantum domain. We cannot obtain a fundamental quantum description of matter by, say, combining the laws of elasticity with the principles of quantum theory; instead, we first
describe the elastic solid as made of billions of atoms and then
provide a quantum mechanical description of these atoms. If
gravity is an emergent phenomenon, then Einstein’s equations
describing the dynamics of the spacetime have a status similar
to the laws of elasticity or gas dynamics. By analogy, we should
not try to quantize gravity by combining Einstein’s theory directly with the principles of quantum theory. Instead, we must
identify the “atoms of spacetime” and provide a quantum description of this structure.
This idea that gravity is an emergent phenomenon has a long
history. It was first introduced by the Russian scientist, A.
Sakharov, in 1968. Another intriguing connection, suggesting a
similarity between the surface properties of black holes and fluid mechanics was investigated by T. Damour, Kip Thorne, W.
Price and their collaborators in the eighties. An attempt to obtain Einstein’s equations from a thermodynamic perspective
was made by T. Jacobson in 1995. Other approaches, very similar in spirit, were developed by G. Volovik and Bei-Lok Hu. My
collaborators and I started on a concrete programme to explore
this idea in 2002. More recently, E. Verlinde has tried to reas-
J a n u a r y 2 0 11
semble many of these ideas from the perspective of string theory. This article will mostly concentrate on the efforts by my collaborators and I. If the emergent picture is correct, most of the
previous attempts to quantize Einstein’s equations were in the
wrong direction and similar to someone attempting to discover
atomic physics by quantizing the laws of elasticity. We will need
a major paradigm shift to make progress.
EMERGENT PHENOMENA
let us now translate these ideas to gravity. A natural length scale
in quantum gravity (called the Planck length, see box on right)
can be built from three fundamental constants in physics: Newton’s gravitational constant, Planck’s constant relevant in quantum theory and the speed of light which plays a crucial role in relativity. The description of spacetime in terms of its metric and
other geometrical properties can be thought of as an emergent
phenomenon valid at scales large compared to the Planck length.
Variables like metric, curvature, etc. used to describe spacetime in Einstein’s theory are analogous to density, velocity, etc.
in fluid mechanics and will have no significance in the microscopic description.
In that case, quantization of the metric itself is not of any use
in unraveling the microscopic structure of spacetime, any more
than quantizing the density and velocity of a fluid will help us to
understand molecular dynamics. Just as the proper description of
molecules of a fluid requires introduction of new degrees of freedom and a theoretical formalism based on quantum mechanics,
the microscopic description of spacetime will require the introduction of new degrees of freedom (“atoms of spacetime”) and a
different theoretical formalism. Since we do not know what these
are, one needs to proceed indirectly, in the same way as nineteenth century physicists, especially the German scientist Ludwig
Boltzmann (1844-1906), did before the atomic nature of matter
was directly observed.
Boltzmann realized that the temperature and heat content
of a fluid arises due to the random motion of discrete microscopic structures which must exist in the fluid. These new degrees of freedom, which we now know are related to the actual
molecules—but were hypothetical objects at the time of Boltzmann—make the fluid capable of storing energy internally and
exchanging it with the surroundings. If the fluid is treated as a
continuum all the way down to the smallest scales, without
any constituent atoms, then it is not possible for it to exhibit
thermal phenomena. Given the fact that fluids can be heated
up and can have a temperature, Boltzmann’s genius was in inferring the existence of the atomic nature of matter, before direct observations revealed these underlying discrete degrees of
freedom. So Boltzmann said, “If you can heat it, it has
microstructure.”
Amazingly enough, we can heat up spacetime and thus apply
Boltzmann’s logic to it.
HOT SPACETIMES
one of the intriguing consequences of Einstein’s theory is the existence of observers who cannot access information from some region of space and time. Just as standing at a seashore you cannot
see beyond the horizon, a spacetime horizon prevents these observers from getting information from the other side. The most
well-known example is in the case of black holes in which the sur-
w w w. s c i a m . c o . i n planck length
Getting A Measure of Gravity
A natural length scale in quantum gravity, called the Planck length, can be
built from three fundamental constants in physics—Newton’s gravitational
constant, Planck’s constant relevant in quantum theory and the speed of
light which plays a crucial role in relativity. This length has a very tiny value
of 10-33 cm. (This notation stands for the number obtained by dividing 1 by
10 repeatedly 33 times; that is, 10−1 = 1/10 = 0.1; 10−2 = 1/(10 × 10) = 0.01 etc. In
decimal notation 10−33 will have 32 zeros followed by 1 after the decimal
point. Similarly, the notation 108, for example, stands for the number
obtained by multiplying 10 eight times.)
To see how tiny the Planck length is, recall that the size of the visible
universe is about 1028 cm which is about 28 factors of ten away from the
everyday scale of few centimeters. So, in the macroscopic sector, these 28
powers of ten contain all of the cosmic phenomena. If we now move in the
microscopic direction we need to travel farther—to 33 factors of ten—before we reach the Planck length at 10−33 cm. Thus Planck length is farther
away in the microscopic direction than the edge of the universe is in the
macroscopic direction.
It is this tiny scale which plays the role similar to intermolecular separation in gas dynamics. In the case of gases, the intermolecular separation
is only about 6 factors of ten smaller than everyday scales but in gravity we
have to go much, much deeper.
face of the black hole prevents the signals from the inside to reach
the outside observer. However, what is not often emphasized in
popular literature, is that horizons are ubiquitous in Einstein’s description. The simplest context in which a horizon comes into
play is in the case of an observer who is moving along a given direction with an acceleration. It turns out that such an observer
has a horizon and cannot receive information from the region beyond it, in a manner very similar to the situation in the case of a
black hole.
Research in the seventies by J. Bekenstein, S. Hawking, P. Davies and W. Unruh led to a tantalizing discovery: observers with
horizons perceive the spacetimes to be hot, endowed with thermodynamical properties. For example, it can be shown that an
observer accelerating through empty space—who sees a horizon
—will feel as though he is sitting inside a microwave oven set at a
particular temperature proportional to his acceleration. (See
Turning Up the Heat). This effect is real in the sense that such an
accelerating observer can use this temperature he notices to heat
up, say, a glass of water.
In addition to temperature one can also attribute another thermodynamic variable called entropy to the horizons. Though it
may be less familiar to some of you than the temperature, it is
probably the most crucial entity in thermodynamics. Roughly
speaking, the entropy measures our lack of information about the
state of the microscopic degrees of the system. When we study the
properties of a gas, say, we cannot measure the positions and
speeds of each of the molecules in the gas. Instead, we describe
the state of the gas in terms of some macroscopic observables like
its volume, temperature—which essentially is a measure of the average kinetic energy of the molecules—and its pressure. There are
a large number of possible distributions of molecules inside the
gas with different speeds which will all result in the same average
S C I E N T I F I C A M E R I C A N I n d i a 33
properties like pressure, volume and temperature. An analogy
might be useful here:
Suppose you are told that on tossing a coin 1,000 times, it
turned up heads 480 times. This kind of macroscopic information
is consistent with several different microscopic distributions of
heads and tails in each individual toss, just keeping the total number of heads at 480. In such a case, we say that there exist several
possible microstates, consistent with what we know about the
macrostate. This lack of precise information about the microstates
(in this case, the precise result of each toss; in the case of a gas, the
precise position and speed of each molecule) is what is encoded in
the entropy. The more microstates are possible with some macroscopic specification, the larger is the entropy.
This connection between lack of information about a system and its entropy turns out to be very general. Because of this
connection, one can also attribute an entropy to the horizons.
In the case of gas, the lack of information is due to our inability
to keep track of billions and billions of molecules and our resorting to an average description. In the case of observers
with a horizon, it is the blocking
of information by the horizon
which leads to the entropy.
In a way, the entropy associated with a horizon turns out to be
more important in the theoretical developments compared to
the temperature of the horizons.
The description of gravity based
on the equations proposed by
Einstein is very natural when
space has three dimensions.
Physicists have, in recent years,
toyed with the idea that the space
could actually have more than
three dimensions with the extra dimensions being very tiny and
compact and hence not detectable in low energy experiments. The
natural generalization of Einstein’s theory to higher dimensions
leads to a class of models known as Lanczos-Lovelock models of
gravity. All these models share the fact that observers who perceive a horizon will attribute thermodynamical variables to the
horizon. It turns out that the temperature attributed by an observer to a particular horizon does not depend on which theory one is
considering; i.e., whether it is Einstein’s theory or Lanczos-Lovelock theory. But the entropy associated with the horizon does contain information about the underlying theory. The entropy happens to be just proportional to the area of the horizon in Einstein’s
theory but is given by a much more complicated relation—first
pointed out by Bob Wald in 1993—in other theories. This is what
makes entropy more important than temperature. Roughly speaking, this is due to the fact that entropy actually contains information about the microstates of the system.
These facts about the horizon were known for several decades
but most people thought of them as rather tantalizing curiosities.
But from the perspective of emergent gravity, they acquire a much
deeper significance. To see this, recall Boltzmann’s dictum: “If you
can heat it, it has microstructure.” It is obvious that spacetimes can
be hot from the perspective of observers who perceive a horizon.
Clearly, there must be microscopic degrees of freedom in the
spacetime which are responsible for this thermal behaviour. The
The connection
between
gravity and
thermodynamics
which started
out as an
analogy in the
seventies has
now become a
physical reality
34 S C I E N T I F I C A M E R I C A N I n d i a
connection between thermodynamics and gravity is not a mathematical curiosity but a physical reality that needs to be taken into
account properly. I started exploring this angle in 2002 and this
program has led to interesting insights.
A THERMODYNAMIC PARADIGM FOR GRAVITY
The starting point was to explore the connection between the
dynamical equations which describe gravity and the thermodynamics of horizons. In a normal thermodynamical system, say,
a gas in a box, changes in the physical parameters like the volume will lead to changes in heat content and internal energy.
The heat content can be related to the entropy and one can
write down a relation between the changes in entropy, volume
and internal energy of the gas. This is usually called the first
law of thermodynamics.
I discovered in 2002 that exactly the same relation holds for
horizons in Einstein’s theory. Moving the horizon by a small
amount turns out to be analogous to changing the volume of a
gas and Einstein’s equations which describe the spacetimes—
with horizons in slightly different positions— reduce exactly to
the first law of thermodynamics (See The Thermodynamics Behind Einstein’s Equations). Since 2002, my group as well as several other groups, have investigated the situation in a wide
class of models more general than Einstein’s theory. In all the
cases, it has been found that the gravitational field equations of
the theory become identical to a thermodynamic law. This suggests that the concept of atoms of spacetime is quite real and
has direct physical content. Around 2004, I realized that, if this
idea is correct, one should be able to directly determine the
density of these microscopic degrees of freedom from macroscopic dynamics itself. Once again, this is closely related to another key fact about the thermodynamics of gases which had
intrigued people before the atomic nature of matter was understood. It turns out that, at a given temperature, each microscopic degree of freedom can store an amount of energy proportional to the temperature. This principle is called equipartition of energy since it says that the total available energy is
divided equally among all the available degrees of freedom.
(There are exceptions to this rule but they are not relevant to
our discussion.) More importantly, the equipartition law gives
a direct link between microscopic and macroscopic physics of
matter. If you know the total thermal energy of a body, and its
temperature, you can immediately calculate the number of microscopic degrees of freedom of the body. If you take one mole
of gas and keep it at what chemists call standard temperature
and pressure, it will contain a definite number of degrees of
freedom. This number is called Avogadro’s number, named after the nineteenth century Italian scientist. The importance of
these ideas lies in the fact that they were developed before one
had direct evidence about the atomic structure of matter. Even
without any direct observational evidence for atoms or molecules, nineteenth century physicists could count the number of
molecules in some amount of gas without knowing quite precisely what they were counting. We are in a similar situation as
regards atoms of spacetime. We now know that a spacetime,
just like a material body, can be hot. Following Boltzmann, it
seems natural to attribute this heat content to microscopic degrees of freedom. But does the law of equipartition hold? If so,
can we use it to estimate the number of microscopic degrees of
J a n u a r y 2 0 11
degrees of freedom
The Thermodynamics Behind Einstein’s Equations
There is a remarkable connection between the first law of thermodynamics
and the laws describing gravity in a wide
class of theories. The connection can be
explained by an analogy.
Suppose, some amount of gas is confined to a box, the volume of which can be
changed by moving the piston. If one lets
the piston move outward due to the pressure of the gas, one can extract some mechanical work from the gas as well as
change the internal energy of the gas. The
gas can also exchange heat
with the surPiston in a
roundings that can be expressed
in terms
displaced
locationof
the entropy change
Piston of the gas. The first law
of thermodynamics relates the changes in
these three quantities: entropy, internal energy and mechanical work.
Let us move on from a box of gas to a
spacetime with a horizon. The location of
the horizon plays a role analogous to the
position of the piston. While you cannot
push around a horizon, you can certainly
Gas made of molecules
consider two different spacetimes with the
Piston in a
displaced location
Piston
Gas made of molecules
Horizon in a
displaced location
HORIZON
Black hole
freedom present in a given region of spacetime?
Remarkably enough, both of these can be achieved. I did
this calculation for Einstein’s theory in 2004 and extended it to
a more general class of theories last year. These results show
that the laws governing gravity in a wide class of models imply
the equipartition of energy among the microscopic degrees of
freedom. What is more, this analysis allows one to compute the
number density of the atoms of spacetime.
When you carry out the analysis you also discover one crucial difference between the gaseous system and gravity. In the
former, the relevant degrees of freedom reside in the bulk volume of the gas. In the case of gravity, the relevant degrees of
freedom corresponding to a region of space reside on the
boundary of that space and not in the bulk. (Physicists call such
systems holographic.) In Einstein’s theory, the surface density
of these degrees of freedom is a constant and hence the total
number of degrees of freedom increases in proportion with the
surface area. In more general theories, the degrees of freedom
still reside on the surface but their density is not a constant. In
fact, given the mathematical expression for entropy density,
one can reconstruct the full theory. It is now obvious that the
proper description of gravity should start from the entropy
density of spacetime—or equivalently, the density of atoms of
spacetime, which is the same thing. This is again in perfect
analogy with the thermodynamic description of matter. One of
the principles of thermodynamics tells you that, in equilibrium, the entropy must be maximum. If the entropy of a system,
w w w. s c i a m . c o . i n horizons at two slightly
different
locations. This disHorizon
in a
displacedagain
location
placement of the horizon
causes changes in
the properties of the spacetime which—in turn—is
governed by the equation describing the gravity.
Remarkably enough, one can prove that this equation reduces
to a form identical to the equation in
HORIZO
N
the case of gas with a piston.
This result was first obtained by me in 2002 in the
simplest context of horizons in Einstein’s theory. Further work by different groups has now established
that this result
is true for a wide class of theories of
Black hole
gravity much more general than Einstein’s theory. In
the general context, the temperature associated with
the horizon is independent of the theory one is studying but the entropy depends crucially on the theory.
Remarkably enough, the thermodynamic description
picks out the correct expression for entropy in each
theory thereby showing that the entropy density associated with a horizon contains the necessary information to reconstruct the underlying theory. This, in
essence, reduces the description of gravity to a thermodynamic one based on entropy density contributed by the microscopic degrees of freedom.
—T.P.
say a gas, is given in terms of other relevant variables then one
can figure out the equilibrium properties of the system by just
maximizing it. Aseem Paranjape and I found in 2007 that exactly the same principle holds for gravity. We could write down
a mathematical expression for the entropy content of a region
of a spacetime and by maximizing it, obtain the equations governing the dynamics of the spacetime in a wide class of models
of gravity including Einstein’s theory. A spacetime obeys Einstein’s equations because the atoms of spacetime maximize the
entropy—just as a gas obeys gas laws because the atoms of gas
maximize its entropy.
The connection between gravity and thermodynamics which
started out as an analogy in the seventies has now become a physical reality. The situation has an exact parallel with the days of
Boltzmann when we had no direct evidence for the atomic nature
of matter but thermodynamics held the key. We have no direct
evidence for the atoms of spacetime but, thanks to the thermodynamic description, we can say a lot about them. n
more to explore
Thermodynamical Aspects of Gravity: New insights, T. Padmanabhan
Reports in Progress of Physics, 73, 046901 (2010) [available online at
arxiv.org/pdf/arXiv:0911.5004]
Black Holes and Time Warps: Einstein’s Outrageous Legacy, Kip S. Thorne ,
W. W. Norton, 1995.
A Journey into Gravity and Spacetime, J.A. Wheeler, Scientific American Library,
W. H. Freeman, 1999.
S C I E N T I F I C A M E R I C A N I n d i a 35