Special Relativity Notes:
1.
Simultaneity:
Assume that the speed of light is constant for all observers. Then we can use this to arrive at a
definition of simultaneity that is not the (merely intuitive) notion of “things that happen at the
same time”, but instead is connected to a real observation (measurement) we can make to
determine if two (distant) events are simultaneous.
Note: We assume that two events can be directly observed to be simultaneous if they happen
at the same place. So I can, for example, synchronize two clocks that are right here with no
trouble at all (signal speed is not an issue when the distances are small enough). The
difficulty about simultaneity arises only for events that are distant from each other. For
example, to synchronize a clock here on earth with a clock orbiting alpha centauri, we could
send a signal to alpha centauri telling them our clock’s reading. But the people receiving the
signal need to decide how long the signal took to get there, and set their clock to our reading
plus the signal-travel time. IF we can’t reduce that signal time to 0, i.e. if there is a finite
maximum signal speed, then there is no way to avoid this problem. And if different “points of
view” disagree on how long the signal took to travel that distance, then they will disagree
about whether the clocks are synchronized at the end.
Einstein’s definition:
Place an observer at the midpoint between the two events. She observes light from both
events, light that has traveled equal distances at equal speeds, and so in equal times. If she
observes that the light from both arrives simultaneously, then she can conclude that the events
were also simultaneous.
A consequence:
However, someone else moving towards one event and away from the other will perceive
light from the first before light from the second, even if she too is at the midpoint of the two.
We can draw this situation on a simple space-time diagram:
O here is an observer who is stationary in the x,t frame; O’ is an observer moving from left to
right in the x,t frame. Both agree that E1 and E2 occurred at equal distances from them. O
sees light arrive from both events simultaneously, and concludes that E1 and E2 were
simultaneous. O’ sees light from E2 before she sees light from E1, and concludes that E2
preceeded E1. Since both are correctly applying Einstein’s definition, both are correct, which
requires that simultaneity be relative to the observer (and, in particular, to her state of
motion).
2.
Time Dilation
Not only do different observers (in motion with respect to each other) disagree about what
events are simultaneous with what other events. They also disagree about measurements of
temporal intervals, i.e. about the time between two events. To see this, we can think about a
kind of (purely theoretical) clock that the principle of the constancy of light’s speed
underwrites—the light clock.
A light clock operates by bouncing light back and forth between two mirrors set some
distance apart, and counting the bounces. Since we have assumed the speed of light is
constant for all observers, such a clock is theoretically perfect.
But if one clock is moving by another, the moving clock will seem to be running slow. We
can see this clearly with the help of yet another diagram:
We can calculate just how much slower the moving clock is running. The distance the clock
travels while the light moves from one mirror to the other is v t, where v is the velocity of
the clock and t is the time the light takes to go from the top mirror to the bottom mirror. By
the Pythagorean theorem,
d2 + (v t)2 = (ct)2
Gathering like terms, we get:
d2 = (c2 – v2) t2
Solving for t, we get
t = d / (c2 – v2).
Recognizing that the time we want to compare t to here is d/c (the time required for light to
travel from top to bottom for the stationary mirror), we divide the top and bottom by c, to get
t = d/c / (1- v2/c2)
If we now write d/c as 1 (adopting this as the unit of time), we get the time dilation equation:
t = 1/(1-v2/c2)
This equation expresses how long a process that takes place in 1 unit of time in the rest frame
will require (from the rest-frame point of view) if it takes place in the moving frame.
An important observation: This effect is symmetrical. Since each frame is entitled to
regard itself as “stationary”, all frames will regard clocks in other frames as running slow.
This is possible only because the different frames are making different comparisons between
the various clocks, each based on the standard of simultaneity for that frame.
3.
Length Contraction
Measuring the length of a moving body can be done in two ways. We can determine the
simultaneous positions of the two ends of the body, and then measure the distance between
those positions. Or we can time how long the body takes to pass; moving at v, if the time
required for the body to pass a given point is t, the length of the body is vt. But we have
just seen that from the point of view of a moving observer to whom the length of the body is
L, our clock will be running slow (requiring 1/(1- v2/c2) seconds for each second it
measures), and so the result of our measurement will be L (1-v2/c2), rather than the “correct”
(rest) length, L. That is, the length of the moving body, as we observe it, will be contracted
(along the direction of its motion) by the factor (1-v2/c2). As v approaches c, this factor
approaches 0, and the apparent length of the body will be reduced to 0. (The same factor
emerges if we use the “simultaneous positions of both ends” approach to measuring the
length.)
4.
The Lorentz Transformations
Gathering these effects together, we can write a group of equations that show us how to
convert the location of events in the coordinates of one frame into a location for the same
events in another frame. These transformations were first proposed by Lorentz, who intended
them to express the effect that travelling through the ether has on measuring devices. His aim
was to explain, by means of this strange “universal” force acting on our measuring rods and
clocks, why it was impossible to detect our motion with respect to the ether. But from
Einstein’s point of view, these same transformations simply reflect the correct way of moving
from one frames’ point of view to anothers’, given that all frames will agree that the speed of
light is c (and in fact, all frames will agree on all the laws of physics). The equations take us
from values for x,y,z and t (locating an event in one frame) to x’,y’,z’ and t’(locating the same
even in a frame moving along the x axis at v):
x’ = (x-vt)
(1- v2/c2)
y’ = y
z’ = z
t’ = (t – vx/c2)
(1- v2/c2)
x-vt, in the first equation, traces the motion of the origin of the moving system as it travels
along the x axis. The factor in the denominator, on the other hand, expresses the contraction
of the x’ “measuring rod” as seen from the x,y,z,t system. In the last equation, the bottom
factor expresses time dilation, while the top captures the shift in simultaneity relations
between the two frames. (Note that v/c 2 is a constant, so this will give us a straight line for
each value of t.) Thus, like the top of the first equation, it sets up how to identify the “0”
point of the t’ time line at different points in space.
5.
Some philosophical reflections
a. Realism and anti-realism about science. This is an ongoing debate in the philosophy of
science. Some philosophers think that science aims to find out the truth about the world.
They are called “scientific realists”. Others hold that the job of science is to systematize the
experiences we have of the world (especially the sorts of experiences that arise from
systematic, careful measurement and observation). Collectively, these philosophers are often
called “instrumentalists”; some prefer “empiricist”. From the first point of view, Einstein has
shown us something very surprising and important, viz. that time and space a)re not separate
entities, but in fact are closely intertwined aspects of one thing, called “space-time”. But from
the second, empiricist, point of view, Einstein has shown something subtly different—he has
shown that when we compare measurements with observers who are moving (at a constant
velocity) with respect to us, we can make their results agree with our own by applying certain
transformations.
In a sense, the empiricists’ way of putting this is more conservative. Empiricists don’t
venture to say what space and time are really like; they only assert that measurements made
by the two groups will be related in certain ways. So they stay closer, in what they assert, to
the actual observations that we make. But the empiricist must draw a clear line, somehow,
between the sentences that express (or could express) observations, and the sentences that say
such (realistic) things as “time and space are not really separable”. And drawing such a line is
a difficult thing to do. The language we use to report our measurements and observations
often includes the terms that appear in the theory we use to explain and predict those
measurements. So we can’t draw the line simply by means of a carefully restricted
observational vocabulary. And other attempts to draw the line (by identifying parts of the
theory’s “world” as observable, and declaring “equivalent” any other theory whose
observable parts have the same structure as those of the first theory) leave us with the
challenge of identifying real theory pairs that have this equivalence but are not simply trivial
variations on a single theme.
On the other hand, the realist can take the language of physicists at face value. But she owes
us an explanation of how the evidence (which is limited to what we can observe) can support
a conclusion that reaches beyond that evidence. We simply cannot observe everything that a
theory says about the universe. But if we conclude that the theory is true, we are claiming
that whatever it says goes, even though we can only check those parts that are observable.
The scientific realist owes us an epistemology that justifies such commitments without
making irrational and/or arbitrary choices the basis for our world view.
b. Rationalism and empiricism. This is an older debate in philosophy, which began early in
the modern era with rationalists like Descartes, Leibniz and Spinoza, and empiricists
including Hobbes, Locke, Berkeley and Hume. The rationalists followed Plato in supposing
that there are certain basic ideas (and truths) that are (in a sense) built into our awareness.
Our minds can intuitively grasp these ideas and truths, and so we can know a priori (i.e.
independently of experience) that (for example) the world’s geometry is Euclidean, or that
every event has a cause. Empiricists claimed instead that all of our ideas come from
experience, and that a priori knowledge was limited to pure logic and mathematics. These
forms of knowledge express necessary relations between ideas, and we can perceive these
relations once experience has furnished us with the relevant ideas.
Empiricists gained a huge advantage with the modern mathematical understanding of
geometry and its relation to physical space. Instead of a leading example of intuitive,
seemingly a priori knowledge that constituted a powerful theory of how our measurements
work out in the physical world, we had now a range of options (different geometries) any one
of which could be made to “fit” our measurements (with the help of universal forces, if
necessary.) Now “theories” that seemed intuitively quite different (Euclidean vs. Riemannian
geometries) turned out to say the very same things about the physical world (if universal
forces were invoked to adjust each to the facts as measured), and what a theory says about the
world turns out to be (arguably) just a matter of what it implies about our observations (i.e.,
for geometry, the measurements).
So we have the standard Empiricist view of today: Pure mathematics is a game of definitions;
axioms are not true of anything, but simply postulated, and theorems are then proved to
follow from those suppositions. This is a priori knowledge, but it tells us nothing about the
world—it simply tells us that if (interpreted in some way) the axioms are true, then the
theorems will be, too. Applied mathematics, on the other hand, is applied to some physical
system that we can measure or observe. If it is true, it must fit all the measurements, and it
cannot be known a priori to be true—we can only test its truth against the measurements (here
some position on universal forces is presumed, since without such a position, we can’t test the
geometry or apply it at all).
But some continue to hold that we have real a priori knowledge, perhaps by means of a
Platonic grasp of the properties that are instantiated in the world. These philosophers often
appeal to our understanding of mathematics and logic—here at least, they claim, we grasp
properties and relations that are not empirical at all, but matters of pure thought. For example,
we said above that in pure mathematics we can prove that the truth of the theorems will
follow for any application in which the axioms (as applied) are in fact true. But even this
(though a matter of logic) is a substantial bit of knowledge. How do we come by it? What is
it that “reveals” the relation of logical consequence to us, if we know it independently of any
experience of the theorem’s truth going along with that of the axioms?
c. Observation and meaning. One important element in Einstein’s reasoning is his insistence
on linking basic terms like “length” and “simultaneity” to actual measurements. This is a
tendency that is tied to the development of positivism, a philosophical doctrine that tried to
sort out questions that are meaningful from the (they thought) meaningless questions of
traditional metaphysics. The key to making this distinction was to require meaningful
questions be answerable in a way that (at least in principle) settled the issue. Since many
questions of traditional metaphysics seemed not to be answerable in this way, they were
rejected as empty and meaningless. For example, consider the hypothesis that the entire
universe, and everything in it, doubles in size every day. Since the changes are proportional,
we would never notice such a doubling (nor would we notice a daily halving of the size of
everything). No measurement or observation could ever support or refute such a claim. So
(say the positivists) it’s actually meaningless.
A wonderful thing about the special theory of relativity is how much mileage Einstein gets out
of insisting that we examine some of our basic physical concepts in this way. Though he does
assume a pretty idealized account of observation (no mistakes, no breakdowns in our
instruments), he still manages to give an account on which length and time and etc. are all
explicitly linked to measurements and observations that could be made. The result is an
account of the world that recognizes explicitly that one (Galilean) frame of reference is as
good as the next for the purpose of arriving at the fundamental laws of physics. The next step
in this process, as we will see, was to figure out how to remove the restriction on Galilean
frames, by invoking “gravity” to compensate for any acceleration the frame of reference is
subject to. The result is a much more abstract (and much more fundamental) characterization
of the basic laws, according to which the same basic laws hold for all frames, not just the
Galilean frames.
So we see that very illuminating (and testable) physics follows from a careful examination of
fundamentals like measurements of distances and time, and a philosophical insistence that the
notion of distance and time cannot be invoked in physics without a clear, principled stand on
how observation (measurement – of a certain idealized kind) settles questions about distance
and time. This is a strong testimony to the importance of thinking carefully about
fundamentals, and the potential for philosophical reflections to play an important role in
physical theory building. As we’ll see in the next section of the course (on quantum
mechanics), worries about the connection between observation and theory have played a
substantial role in the other main branch of 20 th century physics as well.