Download Einstein and Relativity 0.1 Overview 0.2 Discrepancies With

Einstein and Relativity
0.1
Overview
Near the end of the 19th century, there were a number of observations that were inaccurately predicted using
Newton’s theories. These discrepancies gave rise to a new set of physical theories. Here, we will concern
ourselves with the new theories of motion (Special Relativity) and gravity (General Relativity).
0.2
Discrepancies With Newton’s Theories
By the middle of the 19th century, Newton’s laws had been relied upon with little revision for two centuries.
Advances in the fields of electricity, magnetism, and thermal physics left many researchers believing that a
complete description of nature would be available in the coming decades. There were only a few problems
left to be solved.
However, researchers were tackling these problems diligently, facilitated by increasing technology that led
directly to very accurate measurements and observations. These measurements served to make the unsolved
problems worse: Newton’s laws, so accurate until then, were now beginning to show that they were not
correct in all situations. Two situations turned out to be related.
The first of these situations was the perihelion advance of Mercury. In 1870, Leverrier began to work on
discovering the reason for this phenomenon, which can be described as the slow but steady rotation of
Mercury’s orbit. The planets pull on each other, deforming their orbits, but this effect alone could not
account for the observations.
The second of these situations was highlighted in 1887 by an experiment performed by Michelson and Morley.
They measured the relative speed of two light beams travelling in different directions. The theory at that
time predicted that the speed of the Earth through space would add itself to the speed of the light beams,
so beams travelling in different directions would have different speeds. In fact, the two light beams had
identical speeds, indicating that the speed of light was constant and independent of the speed of the source
of light (or the speed of the observer). We use the letter c to represent the speed of light.
0.3
The Postulates of Special Relativity
The result of the Michelson-Morley experiment was correct, and became one of the two axioms upon which
a new theory of motion, Special Relativity, was based. The other axiom was from a process of reasoning
begun several centuries earlier by Galileo. In his book, Dialogues, Galileo observes that a dropped rock will
fall straight down, regardless of whether the rock is held by a person standing still, or a person standing in
the crow’s nest on a moving ship. In other words, the dropping rock can say nothing about the motion of
the ship.
Eventually, through the further work of Galileo, Newton, and others, people came to speak about frames
of reference. A frame of reference is any frame of experience that is moving at a constant speed. We refer
to this unaccelerated frame as an inertial frame. When in a frame of reference, you can only measure
velocities relative to yourself. In other words, only relative motion is measurable, not some absolute motion
defined by a universal sense of being at rest.
Taking the above observations together, we write down the two axioms of Special Relativity:
1. The laws of physics are the same in any inertial (i.e., unaccelerated) frame,
2. all inertial observers measure the same speed of light, c, which is therefore a constant.
0.4
The Theory Special Relativity
These two axioms appear to contradict each other, and yet both are experimentally true, so there must be a
way out of the contradiction. The way out was explained by Albert Einstein in 1905. Einstein concluded that,
in different inertial frames, the observer must measure speeds differently. Speeds are discovered from lengths
and times, so therefore lengths and times must be different for different observers. Further experiment shows
that masses, too, are changed in different inertial frames.
At this point some nomenclature must be introduced. When speaking of an object’s length, mass, and time
interval between events (i.e, the rate of the ticking of a clock), we must distinguish between these quantities
to an observer at rest and an observer in motion. We define the rest mass, M, the rest length, L, and the
rest time interval, T. These are related to the running mass, the running length, and the running time, by
a quantity called gamma, γ. The running mass is M/γ, the running length is Lγ, and the running time
interval is T/γ. Note that both M and T increase (dilate) whereas L decreases (contraction).
v/c
0
0.1
0.3
0.5
0.7
0.9
1
γ
1
0.995
0.954
0.866
0.714
0.436
0
p
What is γ? It is the value of the numerical expression 1 − v 2 /c2 , and can be derived from
the postulates. Looking at the table, you can see that at low speeds, 0.1c, the relativistic
factor γ is very close to 1; as a consequence, Newton’s laws hold true at low speeds. At above
0.3c, the factor γ begins to be measurable and we enter a new realm called the relativistic
regime, where Special Relativity must be used to make accurate predictions. The change
of mass, length, and time are referred to as relativistic effects.
Again looking at the table, you can see what happens when you try to push an object to move at the speed
of light: its mass becomes infinite, its length becomes zero, and time ceases to tick. In other words, reaching
the speed of light is impossible. The speed of light is therefore the upper speed limit of the universe (it’s not
just a good idea, it’s the law!).
The upper-limit idea can be understood intuitively by considering the running mass. You have to push
harder on (i.e., give more energy to) a more massive object to get it to move. If objects get more massive as
they speed up, then it requires ever increasing energy to speed up a fast-moving object. It takes an infinite
amount of energy to accelerate an object to light speed. Therefore, acceleration to the speed of light is not
possible.
Since light travels at the speed of light, then it must have a mass of zero.
How does Special Relativity make the speed of light a constant? The running observer will try to measure the
speed of light by measuring the distance light travels in a given amount of time. The running observer will
measure a shorter distance and longer time than the stationary observer, in such a way that the measured
speed of light will remain constant for both observers.
For fans of science fiction, relativistic effects are the basis of the “tachyon”, a theoretical particle for
which c is the lower speed limit: tachyons travel at faster than light speed, and therefore γ is the square
root of a negative number, a so-called imaginary number. Tachyons therefore have imaginary mass, and
are proposed to move backwards in time, although this last assertion is not supported by the mathematics.
There are two consequences of Special Relativity that I will discuss: mass-energy conservation, and the twin
paradox. The twin paradox will introduce us to General Relativity.
0.5
Mass-Energy Conservation
That mass increases with increasing velocity indicates that mass and velocity are fundamentally related.
Velocity is a manifestation of energy, so one can go further and say that mass and energy are related. In
fact, they are part of a more general concept we now call mass-energy, and we can convert one directly
into the other. You can consider mass to be “condensed” energy.
According to Special Relativity, and supported by experiment, mass (M) and energy (E) are related in direct
proportion to one another,
E ∝ M.
If you want to change the proportional relationship into one of equality, you must multiply in a constant of
proportionality,
E = M × constant.
It turns out that this constant is numerically equivalent to the square of the speed of light,
E = M c2 .
We have arrived at the most famous equation in physics. The best way to understand this equation is to
re-write it as a ratio:
E
= c2 .
M
In other words, any conversion of mass to energy results in a lot of energy. The great amount of energy
contained in a little bit of mass is the driving fact behind attempts to safely harness nuclear fission and
nuclear fusion as energy sources.
What does conservation mean in this context? It means that in any process, the total amount of energy
never changes, even when energy and mass are converted back and forth from one to the other.
0.6
The Twin Paradox
The twin paradox, first developed by Einstein, goes as follows. Suppose you have twin astronauts, one
Earth-bound and one who travels. The traveller performs a loop that leaves the Earth and then returns.
When the twins compare clocks, they see that the travelling clock has lost some time, meaning that it ran
slower in the voyage. In other words, the travelling twin is now younger.
How can this result occur? Special Relativity states that inertial reference frames are equivalent: each twin
should measure the same speed to the other, and both should age at the same rate, but they do not. The
reason is that the travelling twin was not in an inertial reference frame. The traveller sped up and then
slowed down, and therefore experienced acceleration, and therefore was not in an inertial reference frame.
We conclude that acceleration has its own set of relativistic effects.
0.7
General Relativity and Gravity
The fact that acceleration has its own set of relativistic effects was explored by Einstein, who published his
results in 1916 as the General Theory of Relativity.
Acceleration is detectable by experiment. An observer can detect the acceleration of the local frame of
reference; this acceleration manifests itself as a force pushing on the observer. However, gravity also exerts
a push on the observer. The forces due to acceleration and gravity are indistinguishable from each other
through experiment. Note the exact wording here: although the cause of each force is different, the effects
of the force itself are identical.
The identical nature of acceleration and gravity is called the principle of equivalence. If acceleration has
its own relativistic effects, then so does gravity. Therefore, living in an environment with gravity is living in
an inherently relativistic environment.
Living in a relativistic environment has profound implications. Consider an observer who is accelerating.
The path of a beam of light is straight, but to the accelerating observer this beam is constantly moving with
respect to the observer at an ever-different rate while maintaining the same speed. The result is that the
beam appears to curve. Now think about gravity. Nowhere in the universe is free from the force of gravity
(i.e., an acceleration), therefore, all paths are curved, and we must conclude that space itself is curved.
Now we must consider space not as a 3-dimensional object, but rather as a curved 4-dimensional space-time
object.
The new interpretation of gravity, according to General Relativity, is that the presence of mass will curve
the space around it. Objects travelling in this local curvature, also called a gravity field, will deflect from a
straight-line path. Clocks will tick at a slower rate when in a strong gravitational field.
The bending of light due to gravity, a classic demonstration of the accuracy of Einstein’s laws, was first
measured during an eclipse of the Sun in 1919, when the star field behind the Sun was visible. The stars all
showed an apparent shift of position compared to their positions when the Sun was along a different line of
sight.
Finally, we can explain away the perihelion shift of Mercury, the concern which opened this discussion.
Mercury’s elliptical orbit brings it alternately closer to and farther from the Sun. Its travel through regions
of slightly different space-time curvature causes its orbit to shift slightly. Einstein’s theory accounts for the
shift as accurately as we can measure it.
0.8
Final Remarks
The laws of Newton can be shown to be mathematically equivalent to Einstein’s theories at slow speeds and
weak gravity fields. We would expect this result, since Newton’s laws work so well for everyday experience.
However, the relativistic regime is one that we, in the space age, deal with regularly, when placing satellites
in orbit, or interpreting many astronomical observations, or looking for a clean and sustainable source of
power.
A specific example is the programming of a hand-help GPS system, which much take into account General
Relativistic effects on its satellites in order to reach its advertised accuracy. GPS systems are now used every
day for navigation, surveying, and construction.
In 1991, General Relativity was formally adopted as the theoretical basis for space-time reference frames for
the calculation of time and time standards. Clearly, General Relativity has an impact on our daily lives.
0.9
References
Ashby, N., “Relativity and the Global Positioning System”, in Physics Today Stephen G. Benka, ed (College
Park, MD: American Center for Physics) May 2002
Bishop, R., “Time and time scales”, in Observer’s Handbook 2003, Rajiv Gupta ed (Toronto: University of
Toronto Press) 2003
French, A.P., Special Relativity (New York: W.W. Norton & Company) 1968
Will, C.M., Was Einstein Right? (New York: Basic Books, Inc.) 1986