Principles of Relativity
General relativity explains the world in a very different way to our everyday understanding.
This is not to say that it predicts things that actually don’t happen. It predicts exactly what we
know empirically, and seems strange because it stretches beyond the domain of human
experience. One way it does this is in its treatment of the very big, where it shows that a
natural extrapolation of normal events is incorrect. If you imagine trying to draw a graph of
tan(θ) based on results between 0 and 5°, extremely close to a straight line, you can
understand that a theory explaining all phenomena we see at any normal speed, mass or
distance need not follow a seemingly logical extension to huge speeds, masses and
distances. Another problem in the conception of general relativity is that it deals with
dimensions which are largely flexible. Humans live in a world with three largely
interchangeable, perpendicular space dimensions, and a time dimension that is completely
different. General relativity explains the universe in terms of geometrical structures bearing
little resemblance to the familiar physical quantities with which they correlate. They are
impossible to visualise accurately, and even the best representations leave much to be
desired.
The question must then be asked, why is general relativity such an important part of modern
physics? Is it only needed to explain effects at huge magnitudes, such as the precession of
Mercury’s orbit by less than one minute of a degree each century (an effect dwarfed by such
influences as the orbit of Venus) or the bending of light by the sun (by around two arc
seconds and measured for obvious reasons only during solar eclipses)? General relativity in
fact comes from a much more fundamental source than this. The development of principles
of relativity has been one of the most important paths of progress in physics.
Galilean Relativity
Galileo was one of the first scientists to have a truly modern conception of scientists. From
him, centuries of scientists in all disciplines have drawn their purpose, namely to use
experimental results to form theories that can predict results of similar situations in the future.
In order to do this, it is necessary to be able to generalise the results of some experiments so
that future experiments do not need to be set up identically in order to achieve similar results.
Galileo realised that mechanical experiments were not affected by changes in position,
orientation or velocity, provided everything else remained constant. That these criteria are
difficult to achieve is a reason why the principle was not formulated earlier. Rotating a
pendulum 180° about a horizontal axis, for example, will severely affect results of any
experiments that are (somewhat uselessly) performed. Dropping a feather onto the deck of a
stationary ship is unlikely to produce the same results as a similar procedure on a moving
vessel. Nevertheless, the realisation that there were no intrinsic differences between any two
situations in which mechanical experiments may be performed was of immense importance to
development in science’s understanding of the world.
Another of Galileo’s triumphs was the realisation that gravity caused an equal acceleration for
any object in a uniform field, such as on the earth’s surface. It is often claimed that Galileo
dropped balls of different sizes from the leaning tower of Pisa in order to prove this. In fact
this claim cannot be true, because air resistance would be sufficient to ‘disprove’ his theory, at
least in the minds of any critics. While not a part of Galilean relativity, this equivalence is
important in general relativity.
Special Relativity
During the nineteenth century, the development of understanding of the nature of light
inspired many new ideas in physics, particularly in the areas of relativity and quantum
mechanics. Einstein saw the importance of the invariability of mechanical processes under
displacement, rotation or change in velocity (not, however, during the actual act of
acceleration). Electromagnetic experiments, which include all those involving light, were also
invariant under displacement or rotation. It was a natural extension to suggest that
experiments at different velocities would also yield the same results. Einstein made this
suggestion and, guided by Maxwell’s description of light as an electromagnetic wave,
formulated a special theory of relativity. This theory removed the conception of spatial and
temporal dimensions as universal and unchangeable. Instead, the requirement that
observers in non inertial (non accelerated) frames of reference make identical observations
under identical conditions means that observers travelling relative to each other will measure
in the same situation different values of length, time and, depending on definitions used,
mass. The same situation does not imply a uniformity of conditions.
General Relativity
Einstein also realised the importance of the observation of Galileo, Newton and many others
that gravitational mass and inertial mass are exactly equivalent. This means that different
objects will accelerate at the same rate in a uniform gravitational field. No experiment from
Galileo to Einstein, or indeed since Einstein’s time, has proven that this principle is false. In
science, this is sufficient to ensure the truth of this unprovable statement is universally
acknowledged (but not so universally that it would prevent scientists from attempting to
disprove it). The equivalence of gravitational and inertial mass also means that a person,
even with whatever equipment they could require (other than a window) would not be able to
distinguish between being stationary in a gravitational field and accelerating uniformly in the
absence of any gravitational field. There would also be no difference between travelling at a
constant velocity far from gravitational fields and falling freely under the influence of one. This
correlation inspired Einstein’s general theory of relativity.
One of the most important principles in mechanics is that momentum is conserved within a
closed system, one that is not subject to any external forces. While special relativity changes
the mathematical formulation of momentum, it retains this principle. General relativity also
maintains the principle, although it does not treat space in a conventional way. As it does not
assume that coordinates are uniformly perpendicular, it is not possible to add up different
momenta, because they are vectors. In the absence of any normal coordinate system, it is
impossible to add vectors with any meaning. Conservation of momentum under general
relativity therefore relies on the consideration of an infinitesimal volume, the basis of calculus.
In this limit, it can be assumed that coordinate systems behave in a relatively normal way. If
forces are considered to be a flow of momentum, momentum inside this volume will always
be conserved. It is important to note that, because of general relativity’s treatment of space
and time as essentially the same, this does not mean that the amount of momentum inside
the volume remains constant. Momentum which is still there at some moment has ‘flowed
out’ of the volume bounded by the combined dimensions. If the value of any dimension for
some packet of momentum is outside the limits being considered, it has flowed out, and in
this case it flows out along the time dimension.
General relativity relies on the concept of the geodesic. This is the ‘line’ that an object would
follow if no force acted on it. It can be likened to a great circle on the surface of the earth,
which is that path an object will follow if it moves in what appears to be a straight line from its
own perspective. A geodesic can therefore be considered to be a coordinate axis
(remembering that axes are usually straight lines), but one only useful in describing a
particular situation. We can begin with a volume bounded by two geodesics (which cannot
really be thought of as lines or planes, but a visualisation of them as lines can be useful).
Even if these geodesics are initially parallel, we cannot generally say that they remain
parallel. Assuming a resultant change in volume, it must be possible to explain their
separation in terms of density. In this case density is defined as the amount of momentum in
a volume. Using the conservation of momentum, it is possible to formulate an equation
relating the spread of geodesics to the momentum they surround. As momentum is directly
related to mass (and in general relativity, not to velocity), this effectively explains gravity.
Once the equations are known, the influence of mass present at a point on geodesics around
that point, and hence the path of objects moving under its influence, can be determined.
It is possible to determine the equations describing geodesics, and hence space and time,
around a massive object such as a star. These show that at a given distance from the object,
there is a certain area, which is the surface area of a sphere, as expected. Remaining at this
distance is easily possible if one was already following an appropriate geodesic,
corresponding to being in an orbit. Importantly, however, this area is not simply the area of a
sphere with radius equal to the distance from the location to the mass. The area is in fact
larger than should be the case. The closer to the mass one is, the more the apparent radius
of the sphere differs from the distance to the mass. This causes what we observe as
acceleration, as the geodesic corresponding to an object will have a greater change in radius
as it nears the object.
Ideas of General Relativity
General relativity introduces important ideas in how we view the world. The most important is
the conception of space and time, very different to the usual way of considering these
quantities. They are not considered to be three similar and one distinct dimensions
completely describing the location of events, but intertwined quantities that are not uniform. If
they are considered in a relativistic way, however, a complete understanding of the future
events along a geodesic can be formulated. Through the use of these ideas, general relativity
has become one of the most powerful tools for understanding the universe through modern
physics.