There is a fundamental issue in relativity theory. If all the motion is relative how
is it possible to measure the inertia of a body?
The only way this is possible is by measuring it relative to something else. This becomes
more complicated if we consider a spinning reference frame. The earth spins on its axis,
which in turn rotates around the Sun which in turn rotates around the galactic centre. Even the
galaxies move relative to one another. So how can we determine the absolute rotation of an object?
If a ballerina spins, her arms will be pulled outwards. If on the other hand, the ballerina
stands still and the rest of the universe revolves around her, her arms will also be outwards.
If both her and the universe spin at the same rate her arms will not be pulled outwards. Why is this?
How does she communicate to the universe that she is spinning and vice versa? Is there a connection
between all matter and the rest of the universe to communicate the notion of inertia? This is
something that Ernst Mach became fascinated by. In this episode, we will explore Mach's principle
and 2 alternative models to relativity which are able to take into account this principle.
Mach's principle attempts to explain how rotating objects maintain a frame of reference.
The proposition is that the existence of absolute rotation is determined by the
large scale distribution of matter. This is best exemplified by the following anecdote:
You are standing in a field looking at the stars. Your arms are resting freely at your side,
and you see that the distant stars are not moving. Now start spinning. The stars are whirling around
you and your arms are pulled away from your body. Why should your arms be pulled away
when the stars are whirling? Why should they be dangling freely when the stars don't move?
Let's take another example. There are two ways we can measure the Earth's spin about its polar axis.
The first method would be through observation of the rising and setting of stars. The second
method uses a Foucault pendulum. Here the plane gradually rotates around a
vertical axis as the pendulum swings. If you know the latitude of the location of
the pendulum you can calculate the Earth's spin. Both methods give the same answer.
This might not seem so surprising, if we are measuring the same quantity,
we should get the same answer regardless of the method used. But in this example, the two
methods are not measuring the same quantity. The first method measures the Earth's spin
against a background of distant stars, while the second employs the standard Newtonian mechanics
in a spinning frame of reference. In this case, we take note of how laws of motion
get modified when their consequences are measured in a frame of reference spinning
relative to the 'absolute space' in which these laws were first stated by Newton.
Implicit in the assumption that equates the two methods
is the coincidence of absolute space with the background stars.
Mach's principle states that this is not a coincidence. He read something deeper into it,
arguing that the idea of absolute space which Newton used to construct the laws
of motion and arrive at the concept of inertia is somehow intimately related to
the background of distant parts of the universe.
There must be a physical law that relates the
motion of the distant stars to the local inertial frame.
Mach first wrote of this idea in the book The Science of Mechanics where
he criticized Newton's idea of absolute space.
Newton tried to demonstrate that you can always decide if you are rotating
with respect to absolute space by measuring the apparent forces that arise only when an absolute
rotation is performed. Suppose a bucket is filled with water hanging by a rope from
a ceiling and then the rope is given a twist and let go. The bucket will start to rotate.
To an observer in the room, the bucket appears to be spinning like this
However, a fly sitting on the bucket might conclude
that it is at rest and the room is spinning.
If the bucket looked like this
The fly would have concluded that the room was not spinning. Notice the big difference between
these two states is that in the first example, the water curved inwards, rising at the rim. Why
is this curvature there? The fly would conclude that the curvature of the water surface is due
to the centrifugal force that acts on the water mass. Being radially outwards, it would produce
the observed effect on the water. Newton argued that latter case the bucket is at rest relative
to the absolute space, while in the curved water in the bucket example, it is rotating relative to
the absolute space and hence extra inertial forces have to be postulated to explain the
curvature of the water surface. The centrifugal force is the inertial force in this example.
When the bucket is made to rotate, initially the water remains still, but gradually, the walls of
the vessel communicate their motion to the water, making it curve and climb up the borders of the
bucket, due to the centrifugal forces produced by the rotation. This idea demonstrates that
centrifugal forces arise only when the water is in rotation with respect to the absolute space.
According to Mach, the Newtonian discussion was incomplete in the sense that the existence of
the absolute space was postulated arbitrarily and in an abstract manner. He felt that the
universe provided a background reference frame that can be identified with Newton's frame.
Rather than saying that it was accidental that Earth's rotation calculated through
the pendulum and distant stars were the same, Mach took it as proof that the distant parts
of the universe somehow enter into the formulation of local laws of mechanics.
Inertia in an Empty Universe
One way this could happen is by a direct connection between the property of inertia
and the existence of the universal background. In order to understand this point of view, imagine a
single body in an otherwise empty universe. In the absence of any forces, we end up with
F=ma, 0=ma
What does this imply? We could conclude that there
is no acceleration and therefore the body moves with uniform velocity.
There is however no background against which to measure velocity. The lack of any tangible
background for measuring motion suggests that acceleration should be completely indeterminate.
This also leads to the rather remarkable conclusion that mass is also zero.
In other words, the measure of inertia depends on the existence of the background in such a
way that in the absence of the background, the measure vanishes. This aspect introduces
a new feature into mechanics not considered by Newton. The Newtonian view that inertia
is the property of matter has to be augmented to the statement that inertia is the property
of matter as well as of the background provided by the rest of the universe.
This general idea is a consequence of Mach's principle.
Such a Machian viewpoint not only modifies local mechanics,
but it also introduces new elements into cosmology.
Mach's concept was a guiding factor in Einstein's development of the theory
of general relativity. Einstein viewed Mach's principle along the
lines of inertia originates in an interaction between bodies.
Because intuitive notions of distance and time no longer apply in general relativity, the meaning of
'Mach's principle' is even less clear than it is in Newtonian physics and at least 21 formulations
of Mach's principle are possible, some being considered more strongly Machian than others.
A fundamental problem is the very existence of a fixed background, which Einstein described
as 'the fixed stars'. Essentially we wish to separate spacetime into slices of constant time.
By doing this Einstein's equations can be decomposed into one set of equations, which are
satisfied on each slice. There is another set of equations that describe the motion between slices.
Einstein was convinced that a valid theory of gravity
would have to include relativity of inertia.
There are however other theories that are more fully Machian but not without their own problems.
Dennis Sciama's insights In 1953 Dennis W. Sciama,
a Cambridge Physicist, was studying this problem and noted that if the rest of the Universe
determines the inertial frames, it follows that inertia is not an intrinsic property of matter,
but arises as a result of the interaction of matter with the rest of the universe.
This raises the question as to how Newton's laws of motion can be so accurate despite their lack
of reference to the physical properties of the universe. This is one of the reasons for
Mach's principle not been generally accepted. Any theory of inertia that is consistent with
Mach's principle must account for the apparent irrelevance of the properties of the universe.
The observed fact that a gravitational force is locally indistinguishable from an inertial force,
in that each induces the same acceleration in all bodies, suggested to Einstein that it is the
gravitational influences of the whole universe which gives rise to inertia. General relativity
was devised to incorporate this idea, but, as emphasised by Einstein, it failed to do so.
Dennis Sciama used this to give a general argument that there was
a relationship between big G and the large-scale structure of the universe.
The Brans-Dicke Theory of Gravity In order to understand where their idea came
from we must consider how we would compare masses at two different points in spacetime. Masses are
measured in certain units, such as masses of elementary particles, which might themselves be
subject to change. We require an independent unit of mass against which an increase or decrease of
a particle mass can be measured. Such a unit is provided by gravity in the so-called Planck mass:
If we insist on using mass units that are the same everywhere, a change of mass would
tell us that the gravitational constant G is changing. It is equally possible to consider
that Planck's constant or the speed of light is changing. But by assuming only the gravitational
constant changes they followed the principle of least modification of existing theories.
So both special relativity and quantum theory are unaffected by this change.
This is the conclusion Brans and Dicke arrived at in their approach to Mach's principle. They
looked for a framework in which the structure of the universe would give rise to the gravitational
constant G. If G changed this would be looked upon as a Machian consequence of a changing universe.
The Brans-Dicke theory of gravitation is a competitor to Einstein's general theory of
relativity. In it, G is assumed to be variable from place to place and with time. At present,
it is held to be in agreement with observation but it represents a minority viewpoint in physics.
Hoyle-Narlikar Cosmology An alternative approach was put
forward by Fred Hoyle and Jayant Narlikar. I have previously discussed this theory in
more detail in two separate videos that I will link in the description. There is one
important difference between this and the Brans-Dicke as well as general relativity.
This difference lies in the fact that both of those theories are pure field theories, whereas
the one put forward by Hoyle and Narlikar arises from the concept of direct interparticle action.
It is best to show the difference through the description of electromagnetism. Until the
advent of Maxwell's field theory, the electric and magnetic interactions were described as instances
of direct action at a distance between particles. The success of Maxwell's theory established the
field concept in physics at the expense of the concept of action at a distance.
In the action at a distance scenario, the electromagnetic effect from A to B travels
into the future and arrives at B later than it left A. This is called retarded action.
The reaction from B to A travels back along the same route, arriving at A earlier than it left
B. This is called advanced action. The presence of both advanced and retarded effects on an equal
footing makes the problem non-local. In fact, it brings in cosmology. That cosmological boundary
conditions are necessary in the action at a distance framework is seen from the following
simple illustration. Any retarded signal emitted by particle A will get an advanced reaction back
from B. Thus the theory admits advanced signals and appears to violate causality. We can also see
that the signal from b arrives at A at the same time that the original signal left A, no matter
how far away B is! This means electromagnetism ceases to be a local theory and any so-called
local effects must take account of the response of the universe, which consists of reactions
from all such particles B other than A. A correct response can cancel all the causal effects. The
correct response may well depend on the model of the universe. In essence, to produce the correct
response the universe must be able to absorb all electromagnetic signals directed to the future.
But what is the response of the universe?
In the 1930s Dirac demonstrated that when an electric charge A accelerates,
the force of radiative damping that it is subject to can be calculated by evaluating half the
difference of the retarded and the advanced fields of the charge on its own worldline.
In the Maxwell field theory, Dirac's result had remained just a curiosity without proper
understanding as to why the radiative reaction must be determined by the above formula. This in
turn was linked with the more basic question that arises when we discuss electromagnetic fields of
an oscillating system of electric charges. It is customary to choose the retarded solutions of the
Maxwell wave equations to describe these fields, on the grounds of causality. And it is because
of this choice that the system radiates energy and suffers damping. Why do we have a principle
of causality in which causes precede effects, when the basic equations of physics are time-symmetric?
Why do we restrict the solutions to the retarded ones and throw away the advanced ones?
The Wheeler-Feynman theory provides an answer to this. It is formulated in a
time-symmetric manner with advanced solutions on equal footing with the retarded ones.
A typical particle (a) generates a 'direct particle field' defined by the following equation:
The universe as a whole generates a response to these individual fields
of the charges and the 'correct' response from the universe to the motion of (a) is precisely:
It can therefore be shown that for the correct response, the future part of the
universe must be a perfect absorber of all future-directed (retarded) signals,
and the past part of the universe, an imperfect absorber of all past-directed (advanced) signals.
In such a universe if we add this response to the basic time-symmetric
field of (a) we get the net field in the neighbourhood of (a) as
In this way, we get the total effect in the neighbourhood of
a to be a pure retarded one. So a correct response, therefore,
eliminates all advanced effects except those present in the radiation reaction.
It is interesting to note that Narlikar and Hoyle's steady state model generates
the correct response, while all Big Bang Friedmann models fail to do so.
Since Mach's principle suggests action at a distance, Einstein became sceptical
about its validity. Einstein's objections were based on the belief that action at a
distance was supposed to be instantaneous and hence inconsistent with relativity.
In the Hoyle-Narlikar model, they not only incorporate action at a distance but are also
able to successfully describe electrodynamics. Their approach is a rather unusual one.
Let's denote by a,b, the particles in the universe, with ma being the mass of the
ath particle. As implied by Mach, the mass ma is not entirely an intrinsic property of particle a;
it also owes its origin to the background provided by the rest of the universe.
This could be expressed as follows:
This expression means the following. At a typical world point A on the world line of particle a, the
mass acquired by a is the net sum of contributions from all other particles except a in the universe.
The contribution from b to A is given by the scalar function M(b)(A).
The coupling constant Lambda(a) is intrinsic to the particle a. Notice that if a were the only
particle in the universe ma would be zero and we have the Machian conclusion we arrived at earlier.
So in the Brans-Dicke theory, we have a variable constant of gravitation G
and in the Hoyle-Narlikar theory, we have mass which is a result of a communication
with the universe meaning this will change over time. So mass will increase with age.
There is another way of looking at this problem,
a variable speed of light. But that I will leave for a future video.