UNIT
56
Written for students in
the USC Self-Paced Astronomy Courses
Learning Objectives and References are in the Study Guide.
Sample Questions are on the web at http://spastro.physics.sc.edu/
by J. L. Safko
Newton's theory of gravity, first studied in Unit 3, is
intimately related to his concept of space and time. He considered space and
time to be absolute concepts which existed independently of the material
universe. Space was a stage in which the planets and stars existed. As time
passed, the objects in the material universe evolved against the fixed
background of space. Newton also formalized the concept of the inertial frame
(or inertial coordinate system).
A coordinate system (or coordinate frame) is a grid of rods and
clocks at rest with respect to each other that spans a region of space. A
simplified drawing of a coordinate system is shown in Fig. 56-1. Using this
coordinate system we can describe events. Events are things that can be located
at a particular place in space and that occur at a given time. The flashbulb
firing on your camera would be an example of an event. The measurement of an
event is determining the position and time of an event. We also term this
measuring the coordinates of an event.
An inertial frame (or an inertial coordinate frame) is a
coordinate system in which Newton's first law holds. Newton's first law, as
given in Unit 3, is that in the absence of outside forces any body moves with
constant velocity. Any coordinate system moving with constant velocity with
respect to an inertial system is also an inertial system. These inertial frames
were assumed by Newton to be of infinite extent. They covered the entire
universe. According to Newton, once you know any inertial frame, you know them
all, since each differs from another by a constant velocity.
The problem facing Newtonian physics at the end of the 19th
century was that Maxwell's theory of electricity and magnetism contained a
single number for the speed of light. The value of the speed of light would be
the same for all inertial observers. Newton's picture of the universe would
require the speed of light to change if you changed inertial frames. So, there
is a basic disagreement between Newton's world view and Maxwell's equations.
Albert Einstein analyzed this problem and concluded that Maxwell
was right and Newton's laws were an approximation that holds in the limit of
velocities small compared to the speed of light.
Einstein's work changed our ideas on the basic nature of the
universe. He began by asking what the universe would look like if we were to
travel with the speed of light. He also asked how we could build an inertial
frame. Thoughts such as these led him to state
(Positive Statement)
All the laws of physics are the same in every inertial frame.
(Negative Statement)
No test of the laws of physics can in any way distinguish one inertial frame from another inertial frame.
As a consequence of this principle, we are forced to conclude:
The measured value of speed of light is the same for all observers.
Many scientists hold an operational approach to science. They
only want to use concepts that in principle can be observed or constructed. If
we take this approach, the concept of an inertial frame must be re-examined.
How do we operationally define an inertial frame?
Consider a factory that produces many identical measuring rods
(meter or yard sticks) and many identical clocks. By identical we mean the rods
are all the same length to some specified accuracy and the clocks keep the same
time, again to some specified accuracy.
Choose a region of space far from any major masses. We can use
these rods to set up a grid of locations at rest with respect to each other and
we can place clocks at each of these grid points. If we can decide how to
synchronize the clocks, we have constructed a realization of an inertial frame.
This is shown in Fig. 56-1.
Fig. 56-1:
A physical realization of a coordinate system of rods and clocks. The master clock is shown in light color. (after Taylor and Wheeler, Spacetime Physics)
The problem is that we cannot be sure if the rate of our clocks
is independent of their history. That is, the clocks may not be synchronized
after they are placed at the grid points even if they were synchronized when
they were together.
One operational approach is to place all clocks at their grid
points and designate one clock as a master clock. Each of the other clocks is
stopped and set to read noon plus (distance of clock from the master
clock)/(speed of light). When the master clock reads 12 noon, we set off a
flashbulb at the master clock's grid location. Each of the clocks on the grid
is started when the flash from the bulb reaches that clock. This synchronizes the
clocks as shown in Fig. 56-2..
Fig. 56-2:
The master clock in Fig. 56-1 is used to synchronize the other clocks in the coordinate system. If the master clock emits a burst of light at t = 0, the other clocks should start at (distance from master clock)/(speed of light). In the Figure, the clock at coordinates (1, 2, 1) should be set to (distance between 0, 0, 0 and 1, 2, 1) divided by c. That is, it should be set to read 2.4/c and started when the flash is seen.
Note that the operational definition of an inertial frame forces
us to construct a frame of only finite size. An infinite frame would take an
infinite time to construct. That is to say, we have constructed a local
inertial frame and are constrained, by the operational approach, to construct
only local frames
In the Newtonian world view, simultaneity is an absolute
concept. Events which are simultaneous to one inertial observer are
simultaneous to all inertial observers. As we will see, this is no longer true
in the special theory of relativity.
Before discussing simultaneity and other relativistic effects,
we need to carefully distinguish between a measurement and seeing. A measurement
is always made by recording the location and time of events using clocks that
are at rest with respect to each other and rods at rest in the measurement
frame. When you see something, you are receiving light that reaches your
eye at one instant of time. This light may have left its sources at different
times depending upon the distances to the sources.
A measurement will cause moving objects to have shorter lengths,
seeing relativistly moving objects causes surprising distortion. Movies of some
examples
of distortion have been posted on the WEB by Daniel Weiskopf, Theoretical
Astrophysics division, University of Tuebingen
We can discover how the special theory changes our idea of
simultaneity by considering two inertial frames in motion with respect to each
other. We will consider the concrete example of a train traveling at speed,
indicated by the symbol v, with respect to its track bed as shown in Fig.
56-3a. We construct two inertial frames. Frame S is the frame of the track bed
and frame S' is the frame of the train. Each frame has synchronized its clocks as
we previously described.
Fig. 56-3:
(Top Drawing): The three track-side observers, whose clocks are synchronized, measure two lightning strikes to occur simultaneously and coincident with the ends of the train. This means the observer in the center will see both strikes at the same time. She would conclude that the central rider on the train was directly in front of her when the strikes occurred.
(Center Drawing): All three train-based observers have synchronized their clocks in the frame of the train. The observer on the center of the train sees the front lightning strike before he sees the one from the back. He concludes that the front strike occurred before the rear one. The clock readings made by the front and read train-based observers agree with this conclusion.
(Bottom Drawing): The central track-side observer sees both lightning strikes simultaneously, verifying the statements based upon the Top Drawing. The central train-based observer has not yet seen the light from the rear strike, verifying the statements based upon the Center Drawing.
Suppose, to the track-side observer, lightning strikes both ends
of the train when the train is midway past an observer standing by the side of
the track. Directly in front of her is a rider standing at the middle of the
train. To her (the track-side observer) the light from both lightning strikes
reaches her at the same time, so she says the two events were simultaneous.
Assistants of the track side observer standing at (or very near) the lightning
strikes would agree that their synchronized clocks read the same time for each
of the lightning strikes. This is shown in Fig. 56-3b.
As shown in Fig. 56-3c, the observer located at the middle of
the train sees the light from the lightning strike at the front of the train
before he sees the light from the rear-end lightning strike. Likewise, an
assistant on the train at the front-end would read on his clock an earlier time
for the strike at the front of the train than the time measured by a train-observer
near the back of the train.
This gives us the following situation. Each set of inertial
observers have properly synchronized their clocks, but they disagree, in
general, with observers in the other frame.
1. At
the front of the train, when the lightning struck, the train-observer's clock
located there reads an earlier time than the track-side observer's at that
location.
2. When
the observer located at the middle of the train passes the central observer on
the track-side, both agree that their local clocks read zero. This is not
surprising since we arranged this to be true for simplicity.
3. At
the back of the train, when the lightning struck, the train-observer's clock
located there reads a later time than the track-side observer's at that
location.
This means the following two statements are true:
1. Synchronization
of clocks is frame dependent. Different inertial frame observers will disagree
about the proper synchronization.
2. Simultaneity
is a frame dependent concept. Different inertial frame observers will disagree
about the simultaneity of events separated in space.
Associated with this synchronization problem is the question of
measuring lengths. To properly measure the length of a moving object, we must
measure the position of both ends at the same time in our inertial frame.
However, an observer at rest on the moving object would not agree that the
measurements were made at the same time. The observer at rest with respect to
the moving object, using her own clocks, would say that the position of the
front end was measured at an earlier time than the position of the back end. So
both agree that a measurement of length of a moving rod yields a shorter length
than the measurement made in the frame of the rod. This is called the Lorentz
contraction. This contraction is real in any sense of the word as shown in Fig.
56-4.
Fig. 56-4:
The pole-barn paradox(?): An extreme, idealized example of the Lorentz length contraction. A runner is carrying a pole whose rest length is much longer than the barn. Assume the Lorentz length contraction gives an observed length of the rod much shorter than the rest length of the barn. Observers at rest with respect to the barn would determine that the time at which the front end of the rod hits the back of the barn is later than the time at which the back end of the rod enters the barn. Observers at rest with respect to the rod see a rapidly approaching Lorentz contracted barn and conclude the front end of the rod is hit by the back of the barn before the front of the barn passes the back of the rod. For this example the velocity of the runner and rod, the rest length of the rod, and the rest length of the barn must be such that a light signal could not propagate from the front to the back of the rod in the time interval between the two events (front hitting, rear entering). In the language to be developed in section F, the second event is outside of the light cone centered on the first.
Associated with this length contraction is a slowing the rate of
moving clocks. This time dilation is as real as the Lorentz contraction. A
moving clock will appear to measure a smaller time interval than the observer
with respect to whom the clock is moving. Since the interval is smaller, the
moving clock runs slower than clocks in the rest frame of the observer. This is
shown in Fig. 56-5.
Fig. 56-5:
The clocks labeled A and B are two synchronized clocks at rest in our reference frame. The clock labeled A' passes clock A. When this occurs the two clocks (A and A') read the same time. If A' were at rest with respect to A, it runs at the same rate as A. When the clock A', now labeled B', passes clock B, it says less time has elapsed than clock B.
According to the Principle of Relativity, all inertial frames
are equivalent. That means that, not only does the track-side observer measure
the train Lorentz contracted and the train-clocks run slower, but the
train-observer says the track-side observer's rods are Lorentz contracted and
the track-side clocks run slower than the train clocks. Although both observers
agree that this is true it still seems paradoxical at first glance. However the
results become intuitive when we introduce the concepts of spacetime and the
interval.
E. The Interval
How can we resolve this apparent contradiction? What can we say
that all observers will agree upon? Einstein resolved this apparent paradox by
considering space and time as one entity, which we will call spacetime and by
using a different interval rule to find the displacement between two events.
The spacetime of the special theory is not Euclidean. In an
Euclidean space, the separation between two points is given by
distance^{2} = a^{2} + b^{2}
where the quantities are shown in Fig. 56-6a. This is just the
Pythagorean theorem. That theorem relates the two sides of a right triangle to
the hypotenuse of the triangle. It gives the measure of distance in a Euclidean
space.
Fig. 56-6:
- The Euclidean rule for adding distances. The distance is greater than either the side a or the side b.
- In the spacetime of special relativity, the distance is less , in general, than either x or ct. In the case of light x = ct and distance = 0. Since we only have Euclidean paper or screens upon which we make our drawings, don't let the proportions of Figures in spacetime mislead you.
In the special theory of relativity the distance (or
displacement or interval) between two points is given by
interval = square root of[(speed of light)^{2} * (time
interval)^{2} - (space interval)^{2}] . ..........(56-1)
This interval has a minus sign for the spatial portion of the
displacement. This makes the typical interval less than either the spatial or
the time difference. This is the key to why spacetime is non-Euclidean and why
the value of the speed of light is the same when measured by all inertial
observers. For example, the interval between two points connected by light is
zero. This is shown in Fig. 56-6b.
As an example of calculating the interval between two events,
consider the two events.
G, whose coordinates are (ct_{G} =
5.0 m, x_{G} = 3.0 m, y_{G} = 2 m, z_{G} = 0.0 m )
and the event
H, whose coordinates are (ct_{H} =
6.0 m, x_{H} = 3.0 m, y_{H} = 2.5 m, z_{H} = 0.0 m ).
The squared interval is
(ds)^{2} = (c t_{H} - c t_{G})^{2}
- (x_{H} - x_{G})^{2} - (y_{H} - y_{G})^{2}
- (z_{H} - z_{G})^{2}
(ds)^{2} = ( 6.0 m - 5.0 m)^{2} - (3.0 m - 3.0
m)^{2} -(2.5 m - 3.0 m)^{2} - ( 0.0 m - 0.0 m)^{2}
(ds)^{2} = 1.0 m^{2} - 0.0 m^{2} - (-0.5
m)^{2} - 0.0 m^{2} = 0.75 m^{2}.
So,
ds = 0.87 m.
Let A and B be two events as shown in Fig. 56-7. As was stated
in Section A, an event is something that has a well defined location and time.
An example of an event is a flash of light. Let (t1 ,x1, y1, z1) be the
coordinates of event A in a frame we will call S and let (t'1, x'1, y'1, z'1)
be the coordinates for the same event in another frame S'. We will use the
subscript "2" for event B. We use the letter "d" to represent
difference. Then in frame S the separation of the coordinates between the
events B and A are given by
dx = x_{2} - x_{1}, dy = y_{2} - y_{1},
dz = z_{2} - z_{1}, and dt = t_{2}-t_{1},
likewise in S' we have
dx' = x'_{2} - x'_{1}, dy' = y'_{2} - y'_{1},
dz' = z'_{2} - z'_{1}, and dt' = t'_{2}-t'_{1},
The interval separating A and B is given the symbol ds.
Einstein's rules give the square of the interval, ds^{2},
in frame S as
(ds)^{2} = (cdt)^{2} - (dx)^{2} - (dy)^{2}
- (dz)^{2},
while in the frame S' the square of the interval is
(ds)^{2} = (cdt')^{2} - (dx')^{2} -
(dy')^{2} - (dz')^{2},
where c is the speed of light.
Fig. 56-7:
- Two events, A and B, shown in spacetime. The positioning of the events in space and time depends upon the coordinate frame used.
- The two events described in frame S.
- The two events described in frame S'.
Since both of these frames describe the same spacetime, Einstein
proposed that the interval between two events should be the same for all
inertial observers. This means that
(ds)^{2} = (ds')^{2}
......................(56-2)
An observer views a clock at rest in his own frame. The interval
is purely time, no space motion occurs. The same observer notes that a moving
clock runs slower, so the elapsed time is larger than clocks at rest. But the
moving clock covers some space also. When we calculate the interval according
to the preceding rules, we find the spacetime interval is the same.
There is no contradiction. There is a four-dimensional
spacetime. Each inertial frame corresponds to a different slicing of that four
dimensional universe into a three dimensional space and a one dimensional time.
Each different slice is the natural choice for that inertial frame. None can be
said to be better or more correct.
As we inspect the expression for the interval, we see that (c
dt) is treated the same as or dy or dz. That is, since c = the speed of light =
constant, we can consider c to be a conversion factor that converts our usual
units of time to the same units as we use for space. That is, cdt measures the
time interval in units of length. If we consider c to be a conversion factor,
we should not be surprised that it is the same to all observers.
The interval squared, (ds)^{2}, can be positive or
negative or even zero. Whenever the displacement in time (c dt) is greater than
the displacement in space, the square of the interval is positive. Consider a
real body. In the rest frame of the body there is no space displacement, there
is only time displacement. The square of the interval is positive. So, in any
possible inertial frame, the velocity of the body is less than c. No object can
be accelerated to a speed greater than that of light, nor can a real body be
observed moving faster than or even as fast as light.
Consider the case where the square of the interval is zero. This
corresponds to a light path. That is, the displacement in space, dx is equal to
the time displacement c dt.
If we plot "ct" vertically and two of the space
dimensions horizontally, the path of light originating at any point is cone
shaped. We refer to this shape as the light cone. The third space
dimension is suppressed since we can't make a drawing showing it. This is shown
in Fig. 56-8 where we consider intervals measured from the origin to another
event.
Fig. 56-8:
The future (upwards) and past (downwards) light cone passing through x = ct = 0. Event A lies to the future of the origin (x = ct = 0). This means that it is possible to reach event A from the origin at a speed which does not exceed light. Event B lies in the past light cone which means it is possible to go from B to the origin without exceeding the speed of light. Events D and E lie in the elsewhen; they can have no effect on the origin, nor can any action at the origin reach them. There exists a coordinate frame in which D and E are simultaneous.
Imagine the light cone extended towards the past. This extended
cone also satisfies (ds)^{2} = 0, where the upper point is taken as the
origin. The light cone provides an invariant division of spacetime. By
invariant we mean that all inertial observers agree as to which events are in
the upper half cone, the lower half cone and outside the cone. There is the past
and the future where (ds)^{2} > 0, and what has been termed
the elsewhen where (ds)^{2} < 0. It is called the elsewhen
since the events in it are at a different spatial location from the origin and
they may be either at a later time, the same time, or simultaneous with the
origin depending upon the inertial frame used. These terms are also very
descriptive in a causal sense.
Consider an event A which lies in the future as shown in
Fig. 56-8. As shown event A occurs later than t = 0. Since A is
inside the light cone from the origin (t = 0, x = y = 0), the interval
connecting these points is timelike. This statement is true for all observers.
This means if for other reasons we think that some event at the origin causes A,
all observers could agree. Event B which lies in the past also satisifies
this property, as well as being able (potentially) to affect the apex of the
cone (the here-and-now). Events D and E, which lie in the
elsewhen, do not have this property. Depending upon the inertial frame, either
may be said to occur after the other or they may be made to occur at the same
time (simultaneously). These events can have no causal relation.
When discussing an object from two different coordinate frames
we often take one of these frames to be at rest with respect to the object. The
coordinates in that frame are called the proper (or rest) coordinates.
So, we speak of proper time and proper space coordinates. The time in
some other coordinate frame is called coordinate (or laboratory) time and the
space measurements are called coordinate (or laboratory) measures as
illustrated in Table 56-1.
Table 56-1: Proper and Laboratory Frames
Name of frame |
Property of
object under study |
proper or rest |
at rest in this frame |
laboratory or
coordinate |
moving with some velocity not equal 0 |
G. The Twin Effect
One of the interesting predictions of the special theory of
relativity is the twin effect. Consider two twins who were born on Earth. One
of the twins gets in a spaceship which quickly accelerates to a large velocity
(say 0.9 the speed of light for this example) relative to the Earth. This twin
travels for a period of proper time, say 10 years, quickly turns around and
returns to Earth where he deccelerates and stops. To the traveling twin only 20
years have passed, while the twin who stayed on Earth experiences a much larger
passage of time. This is illustrated in Fig. 56-9. The prediction of relativity
is that 46 years will have passed to the twin who stayed at rest on the Earth.
Fig. 56-9:
- One person remaining on Earth as his identical twin leaves on a trip at a speed near light speed.
- When the traveling twin returns, he has aged much less than the one remaining at home on the Earth.
Fig. 56-10:
Spacetime path of the two twins as drawn by the twin remaining on Earth. The elapsed time to the traveling twin (his proper time) is less since the length of the bent path is less than the length of the straight line (x' = constant). Please review the comments on the title for Fig. 56-6b.
Some authors have referred to this as a paradox. They argue that
relativity says all observers are equivalent so we could have considered the
twin in the spaceship to have stayed at rest and the Earth to have moved away
and come back to the spaceship. This argument is incorrect. Only the Earth twin
stayed in the same inertial frame. The spaceship twin felt acceleration upon
leaving the Earth, at the turnaround point, and when stopping at the end. We
assume the accelerations themselves did not affect the clocks, they only serve
to show that a change of coordinate frames occurred to the spaceship twin. That
is, there is no single inertial frame for the twin in the spaceship, while
there is a single frame for the Earth twin. The two twins are not equivalent.
The invariance of the interval can be used to calculate the
predictions for the twin effect and other relativistic questions. For our
example, in the outward path the spaceship twin traveled 10 years, by his
clock, and did not move in his coordinate frame (proper frame). According to
the Earth twin, the space ship traveled a distance dx' = 0.9 ¥ c ¥ dt', where
dt' is the elapsed time according to clocks at rest with respect to Earth. See
Fig. 56-10.
The invariance of the interval says that (ds)^{2} =
(ds')^{2} , where the unprimed frame is used for the spaceship twin and
the primed is the Earth based frame. In the spaceship frame
(ds)^{2}= (c dt)^{2} - (dx)^{2} - (dy)^{2}
- (dz)^{2} = c^{2}(10 years)^{2},
where the last equality is true since the spaceship does not
move in its own rest frame.
In the Earth frame
(ds')^{2} = (c dt')^{2} - (dx')^{2} -
(dy')^{2} - (dz')^{2},
=(cdt')^{2} - (0.9 c dt')^{2} = c^{2}(1-(0.9)^{2})(dt')^{2}
= 0.19 c^{2}(dt')^{2}.
Equating the intervals and removing the common factors of
"c" gives
(10 years)^{2} = 0.19 (dt')^{2}, or dt' = 23
years.
Table 56-2: The Twins
Twin |
Age when trip
started |
Age at turn
around |
Age when traveler
returns |
Earth-based |
20 years |
43 years |
66 years |
Spaceship-based |
20 years |
30 years |
40 years |
The same argument can be applied to the return trip. So, while
20 years elapsed to the spaceship twin, 46 years elapsed to the Earth twin.
These numbers are summarized in Table 56-2. This suggests that you could travel
across the universe in your lifetime, if you could travel sufficiently close to
the speed of light. However, at these speeds, the almost empty space between
the stars would be like a brick wall to the spaceship. Assuming that this
problem can be solved, the universe is ours. However, we won't be able to
return to tell of our travels, since all the people we know will be long dead
when we get back. That takes a lot of the fun out of travel.
The final aspect of the special theory of relativity that we
will consider is Einstein's mass-energy relation. In the special theory of relativity
the separate expressions for energy and momentum are combined into one quantity
called energy and momentum. Included with energy is the energy of existence, or
the mass.
As shown in Fig. 56-11, Einstein considered light emitted inside
one end of a hollow closed tube and absorbed at the other end. His analysis led
him to conclude that mass and energy were intimately related by the equation
E = m c^{2},
where "m" is a mass, "c" is the speed of
light, and "E" is the energy equivalence. In SI units, m is in
kilograms, c in meters per second and E is in Joules. This relation is the
basis of our understanding of the energy source of stars.
Fig. 56-11:
- Light is emitted from one end of a closed box. Since light has momentum, the box begins to move in the -x direction.
- At a time t = L/c, the light is absorbed by the other end of the box. The box then stops moving with a displacement -x. Since the system is isolated, the center of mass of the box, before and after the light travels, must remain fixed. Any wave carries a momentum given by(energy of wave)/(speed of wave). Using this and the assumption that the center of mass of the box does not move leads, after some mathematical analysis, to E = mc^{2}, where m is the mass equivalent of the light whose energy is E. So, an amount of matter, m, was converted to light of energy E and then converted back to the same amount of matter.
I. Astrophysical
Applications
The Special Theory has a number of astrophysical and
cosmological implications. Among there are an understanding of how stars
generate their energy and restrictions on our ability to travel or transfer
information. Many of the astrophysical and cosmological consequences are
strongly dependent upon gravity and thus must await out discussion of the
General Theory of Relativity in Unit 57. In this section we will discuss a few
of the results which do not depend upon the General Theory.
The Einstein mass-energy relation (discussed in Section H)
explained how the stars could generate their energy for long periods of time.
Geologists had arrived an age for the Earth of over 4 billion years while
astronomers could not explain a hot Sun for that length. The understanding of
the mass-energy relation and nuclear fusion removed this apparent contradiction
since the process of fusion would allow the Sun to radiate energy for a time
longer than the estimated age of the Earth.
An interesting astronomical result is that, although speeds of
material objects can not exceed that of light, some speeds can appear to exceed
light. This is a result of our inability to make proper velocity measurements.
This is because we are confined to the Earth and can only deduce a velocity
from what we see rather than by a measurement as defined in Section C of this
material.
Fig. 56-12:
A drawing showing the motion of a bright knot in the jet emitted from quasar 3C273 between 1977 and 1980. These drawing are based upon the original radio data.
A prime example of this is the quasar 3C273. This quasar has
apparently ejected a bright knot of plasma. A simplified sketch of the observed
quasar and the knot at five different dates is shown in Fig. 56-12. Based upon
our estimates of the distance to 3C273, these pictures would seem to imply that
the knot is traveling faster than the speed of light. We can show that this
conclusion is an artifact of the photos, which are not proper measurements as
we discussed in Section C. Since we are not doing the full mathematical
analysis in this material, we can only give the relativistic results.
Fig. 56-13:
A knot of material moving at a relativistic velocity is emitted from a quasar. Two photographs are taken on Earth. The first photo is when the knot is at position "1" and the second when the knot is at position "2".
Consider a simpler example where the source is at rest with
respect to the Earth as shown in Fig. 56-13. The source emits a could moving at
a relativistic velocity, v, at an angle of theta relative to the
source-Earth direction. We take two pictures when the emitted cloud is at
points "1" and "2" in Fig. 56-13. These bursts are
separated by a time Dt (Greek Delta in the illustration) as measured locally by
a coordinate system (clocks and rods) at rest with respect to the Earth. This time,
_t is not what we measure. What we measure is the time interval, _tobs, which
is the time interval between the arrival at the Earth of the two light
emissions. Using the invariance of the interval and c for the speed of light,
these two times are related by
Dt_{obs} = Dt (1 - (v/c) cos[theta] )
.
The photographs do not show the motion towards us, only the
motion transverse to the source-Earth line. We call this distance Dx_{obs}.
The observed transverse velocity, v_{obs}, is given by
v_{obs} = = =
For small theta, v_{obs}, can be very large. The
observed velocity is a maximum when cos theta = v/c for which
v_{obs,max} =
So the observed velocity can be larger than the speed of light.
J. Philosophical
Implications
The restriction of material bodies to speeds less than light
also has great consequences. As we discussed in Section F, starting at an
initial event, the light cone separates possibly causally related regions of
spacetime from those regions which cannot be causally related to the initial
event. This means that if you could travel faster than light, the time order of
apparently causally related events could be reversed for you. You could also
interfere with event which have already occurred to another observer.
The Special Theory avoids these problems since accelerations can
not move a material body to speeds exceeding the speed of light. To accelerate
a body of mass m to the speed of light would take more than (m c^{2})
worth of energy (where c is the speed of light) if the acceleration forces are
provided by the body. Likewise, according to the Special Theory, a fixed
external force produces less and less acceleration as the speed of the
accelerated body approaches the speed of light.
An out may exist in the General Theory (the topic of Unit 57) if
the universe has strange topological connections. Even then there are causal
problems. The only possible out would be if the universe is predetermined and
our ideas of cause and effect are only illusions. That is, without causality,
there is no logical reason to forbid time lines that fold back on themselves.
But is science still possible? I leave these ideas for you to investigate.
An interesting astronomical result is that, although speeds of
material objects can not exceed that of light, some speeds can appear to exceed
light. This is a result of our inability to make proper velocity measurements.
This is because we are confined to the Earth and can only deduce a velocity
from what we see rather than by a measurement as defined in Section C of this
material.
K. Why the special theory
is not the end of it.
Consider yourself in an elevator. You cannot see outside, so you
must determine the nature of the surrounding universe by local experiments. You
let go of a coin and it falls to the bottom of the elevator. Aha!, you say, I
am at rest on Earth. But, you could be in a spaceship that is accelerating and
far from any other object. This is shown in Fig. 56-12.
Fig. 56-14:
Principle of Equivalence I.
Locally being at rest on the Earth's surface is equivalent to being uniformly accelerated in a rocket ship.
Consider the opposite case. You float from the floor and the
coin does not fall when you release it. Aha!, you say again, I am in space far
from any other body. But, you could be freely falling towards the Earth as
shown in Fig. 56-13.
Fig. 56-15:
Principle of Equivalence II.
Locally being in space far from any gravitating bodies is the same as freely falling towards the Earth.
We see that gravity is different than other forces. You can make
gravity completely disappear in small regions by freely falling. This means
that a free fall frame seems to be a perfectly good inertial frame. The only
way we can detect the difference is to look for tidal forces which arise if the
gravitational field is not perfectly uniform. But for any reasonable
gravitational field we can always make the region we consider (our elevator in
this case) small enough so we cannot detect the tidal forces. Another way of
saying this is:
"For sufficiently small regions, the special theory of
relativity is correct!! But, for larger regions we need a more general
theory"
The general theory of relativity was developed by Einstein to
properly incorporate gravity into physics. How this was done and the surprising
consequences will be the subject of Unit 57.
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