-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathspacetime.html
More file actions
492 lines (478 loc) · 26.9 KB
/
spacetime.html
File metadata and controls
492 lines (478 loc) · 26.9 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
<!DOCTYPE html>
<html lang="en" xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>Special Relativity</title>
<meta name="description"
content="How light connects time and space in spacetime geometry.">
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<script>MathJax = {tex: { inlineMath: [["$", "$"], ["\\(", "\\)"]] }};
</script>
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"
id="MathJax-script">
</script>
<style>
div {
-moz-box-sizing: border-box;
box-sizing: border-box;
}
.textcolumn {
padding-left: 1em;
padding-right: 1em;
min-width: 22em;
max-width: 52em;
margin: 0 auto;
}
.halfwide {
width: 60%;
float: right;
}
.clickable {
cursor: pointer;
}
.backlit {
background-color: #d8d8d8;
}
svg {
background-color: #ffffff;
stroke-linecap: round;
stroke-linejoin: round;
width: 100%;
height: 100%;
}
.belowsvg {
font: italic bold large serif;
padding-top: 0px;
margin-top: 0px;
margin-left: auto;
margin-right: auto;
text-align: center;
}
.svgtxt {
font: bold 0.45px sans-serif;
pointer-events: none;
}
@media only screen and (max-width: 620px) {
.halfwide {
width: 100%;
}
}
</style>
</head>
<body>
<div class="textcolumn">
<h1 style="text-align: center;">
<a href="index.html">Special Relativity</a></h1>
<h2>Connecting place and time</h2>
<p>Motion defines time. We have progressed from the spin of the
earth to the vibration of cesium atoms, but we have always
measured time by movement.</p>
<p>We envision motion as a sequence of snapshots: First the runners
are lined up at the starting line, next all have advanced a few
yards, and so on; successive snapshots show the relative positions
changing as some drop back while others move forward, until a
final snapshot shows their order at the finish line. Each of
those snapshots shows where the runners are in space, and each
such configuration marks an instant of time. We can make these
ideas quantitative by measuring positions in each snapshot with a
ruler, or better yet, by placing a tape measure beside the track
so it is visible in each frame. If we also place a clock beside
the tape measure, we can reduce the race to the succession of
position and time coordinates - read from tape and clock
respectively - for every runner.</p>
<p>
</p>
<hr/>
<p>Einstein published his special theory of relativity in two papers
in 1905. The first, "On the Electrodynamics of Moving Bodies,"
suggests a radical physical interpretation - now called spacetime
- for the Lorentz transformation which was known to preserve the
structure of Maxwell's equations. The second, "Does the Inertia
of a Body Depend Upon Its Energy Content?" introduces his iconic
equation $E=mc^2,$ which is to Einstein what $F=ma$ is to Newton.
Astonishingly, the same 1905 volume of Annalen Der Physik contains
two additional papers by Einstein which set the stage for even
greater upheavals in physics than the two on special relativity
(his 1921 Nobel prize is for one of those, not relativity), but
that is a tale for another time. Here and now we will explain the
bare essentials Einstein's theory of special relativity. Like
Einstein, we split our explanation in two parts.</p>
<p>First, we introduce the concept of spacetime coordinates. The
centerpiece of special relativity is the relationship between
coordinate systems built by observers in relative motion
(hence <em>relativity</em>), called Lorentz transformation. We
restrict our discussion to two dimensions - one time and the other
position along a line - and explain the similarities and
differences between this spacetime plane and the familiar
space-space plane of Euclidean geometry. Einstein derived the
surprising properties of high speed motion by carefully analyzing
the question, "What does it mean for two events at different
places to happen at the same time?" His answer explains why
nothing can move faster than the speed of light, by distinguishing
timelike from spacelike directions in the spacetime plane.</p>
<p>Second, we consider collisions between particles or "point
masses" in light of our analysis of high speed motion. When a
moving basketball collides with a stationary tennis ball, the
tennis ball takes off at just under twice the speed of the much
more massive basketball. But this Newtonian analysis of the
collision cannot be correct for very high speed collisions: If a
proton moving at, say, three quarters the speed of light hits a
stationary electron head on, the electron cannot fly away at twice
the speed of the much more massive proton, because, as we found,
nothing can move so fast. The resolution of this conundrum is
$E=mc^2$ - the recognition that energy, in this case kinetic
energy, has inertia and momentum.</p>
<p>The abstract logic of these arguments obscures the most important
fact about Einstein's theory of relativity: Einstein invented
relativity to explain an experimental fact which subtly disagreed
with Newton's laws - Newton's concept of absolute time fails to
explain the results of the Michaelson-Morely experiment. You
cannot make sense of high speed motion using Newtonian mechanics;
it slowly stops working as things move faster and faster. We are
talking about <em>very</em> high speeds - earth orbits the sun
at <span style="white-space: nowrap;">30 km/s</span>, but this is
ten thousand times slower than light and results in negligible
errors in Newtonian mechanics.
</p>
<p>Special relativity is a shining example of the indispensible
roles of both theory and experiment in science: You cannot
interpret what you see without a theory or model, but you cannot
know what questions to ask - let alone what answers to expect -
without careful observation and experiment. Most people associate
science, especially physics, with theoretical physics. This is
natural: Lab courses are adjuncts to lecture courses, where we are
taught how to think, what to expect, and how to argue. And yet
when the going gets tough, experimental science takes over.
Theory always follows experiment, not the other way around. It is
no accident that Newton, author of the foundational laws of
physics, was also one of the greatest experimental physicists in
history.</p>
<p>Special relativity rejects absolute time, the assumption that all
observers will measure the same time interval between any two
events. Newton well knew that his laws of motion are based on the
existence of absolute time - he says so in the first few pages of
his Principia. Einstein pointed out in 1905, 18 years after the
Michaelson-Morely experiment and 218 years after Principia was
published, that the simplest way to understand that experiment is
to reject the concept of absolute time. The real world does not
behave that way. And yet, the only response to an anomalous
experiment is a new theory, for without a model that does explain
the new result, we have no way to use our discovery. So Einstein,
no experimentalist, and his theory get all the popular credit for
the advance when in fact experiment always leads the way.
Einstein was a brilliant theorist, deserving all his personal
credit, but his authority extends only as far as experiment, not
theory, dictates.</p>
<p>The history of relativity reminds us that even the most renowned
experts are slaves to the theories they have built - Newton,
Gauss, Lagrange, Hamilton, and Maxwell all understood very well
that their theories assume the existence of absolute time. In
1905 Einstein pointed out that the Michaelson-Morely experiment of
1887 provides solid evidence against absolute time. There's no
way anyone could have guessed that outcome. Indeed, the whole
experiment was designed to test a prediction of Maxwell's
understanding of how light propagates through apparently empty
space. For twenty years, eminent scientists such as Poincare
concocted complicated schemes to salvage absolute time and
still explain the Michaelson-Morely findings.
</p>
<p>Why is the speed of light $c$ special? Because light is a wave
that can propagate through a vacuum - completely empty space.
Other waves require a medium, a material for the wave to vibrate,
and the wave speed is a property of that material. For example,
the compressibility and density of air determine the speed of
sound. Maxwell and everyone else before Einstein assumed that
outer space must not be empty precisely because light crosses it.
But it is. Exhaustive efforts to determine the speed of the earth
through this "luminiferous ether" in the late nineteenth century
failed; special relativity is how we have come to understand these
experiments. Since the medium for light is empty space, the speed
of light is a property of space itself, the limiting speed between
any cause and its effects. Light is just an example of a
phenomenon which moves through vacuum at the speed limit for cause
and effect that is built into space and time.</p>
<h2>Spacetime diagrams</h2>
<p>We begin with some spacetime jargon. A point in spacetime is
called an event (supposing <em>something</em> must be happening
everywhere all the time). We will plot time upwards and position
rightwards, which is the usual convention for a spacetime diagram.
Any object will have some position at every time, which is a
(maybe curved) line on our spacetime diagram. This is called the
world line of the object. A stationary object has a straight
vertical world line, and an object moving at a constant speed has
a straight inclined world line. Two objects moving at the same
speed have parallel world lines; otherwise world lines intersect
at the event where objects either collide or pass.</p>
<blockquote>[Drawing with two or three pre-drawn world lines and one
you draw yourself. A line below the diagram shows the line in
space; when you begin to drag your point, the spacetime diagram
begins to unfold from the bottom, stopping when you
release.]</blockquote>
<p>You can add your own world line to this spacetime diagram by
dragging the round marker on the line below the figure to the left
or right. When you press down, the diagram above the line records
your dragging movements by scrolling upward at a constant rate.
The square markers move in a predetermined way while you move the
round marker however you please. The slower you move your marker,
the more vertical its world line; faster motion corresponds to a
more horizontal world line.</p>
<p>We have simply asserted that we plot time vertically and position
horizontally. By this, we mean that we plot the locus of events
at a fixed position in space as a vertical line, and the locus of
events at some fixed instant in time as a horizontal line. What
constitutes a "fixed position" obviously depends on the state of
motion of the observer: A moving observer will regard anything
moving with the same velocity to have a "fixed position" relative
to themselves. Thus, our spacetime diagram must always adopt the
point of view of an observer with some particular state of motion
which we arbitrarily call "stationary". In spacetime, the point
of view of an observer comprises not only their location, but also
their velocity.</p>
<h2>Synchronizing clocks relates directions in spacetime</h2>
<p>Since horizontal lines in our diagram are supposed to represent
all events which happen simultaneously, we need a procedure for
synchronizing any pair of stationary clocks (which have vertical
world lines) in order to know where to plot events occuring at the
clock positions. Without moving the clocks, the only way to
synchronize them is to send a signal from one to the other, say
from left to right. By subtracting the signal transit time from
its arrival time at the right clock, we find the time on the right
clock corresponding to the departure time of the signal from the
left clock. Assuming we can send signals through the same channel
(wire or fiber optic or just empty space) in both directions at
the same speed, the one way transit time will be half of the round
trip travel time. The left clock, say, can time the round trip
whether we have synchronized the clocks or not.
</p>
<h2>Symmetry between observers relates scales in spacetime</h2>
<p>In ordinary plane geometry, we begin constructing a system of
coordinates by selecting an origin and a line we will use as
reference direction. Euclid then tells us how to construct
parallel and perpendicular lines we can use to build a coordinate
grid. Parallel lines involve the operation of translation, which
amounts to shifting our origin point, and which we can imagine in
the spacetime plane just as easily as in the Euclidean plane.
However, Euclid's construction of perpendicular lines makes no
sense in a spacetime plane; we need to invent a new procedure to
define the analog of perpendicularity in order to construct a
spacetime coordinate grid.</p>
<p>The key to creating a coordinate system in the spacetime plane is
synchronizing two identical clocks which are a substantial
distance apart. The next figure shows you one way to do that.
The gray lines are inclined at the speed of light: A light flash
traveling to the right will be parallel to the right-sloping
lines, while a leftward traveling flash will be parallel to the
left-sloping lines. Light moves the same speed in either
direction, so the slopes are equal and opposite. The parallel
blue lines are the world lines of two clocks. A light flash
begins at the left clock, reflects off the right clock, and
returns to the left clock; we note start and stop times of this
round trip on the left clock, and the reflection time on the right
clock. To synchronize the clocks, subtract half the round trip
time (as measured by the left clock) from the reflection time on
the right clock to find the right clock time simultaneous with the
start time on the left clock.</p>
<blockquote>[Gray diagonal grid with adjustable slope in background.
Two parallel clock world lines in blue, initially vertical. A
dark gray flash of light world line starts at left clock, reflects
from the right and returns to the left. The left clock world line
is bold from start to stop, and a gray line runs from the midpoint
on the left world line to the reflection point on the right, with
a bold half-length extending backwards to that point. A blue tick
marks the start time on the left world line, and a similar blue
tick sits at an arbitrary point on the right. You can drag that
tick to adjust the origin of the right clock; when it reaches the
base of the bold half-length section, a light blue grid appears,
with the full unit square bold. A blue dot at the top of the left
clock line drags to adjust the velocity of the clocks. A blue dot
at the top of the right world line drags to adjust the spacing
between clocks. A gray dot at the upper right of the gray grid
drags to adjust the slope of the speed of light
grid.]</blockquote>
<ol>
<li>The blue ticks mark the zero time for the left and right
clocks. Drag the right tick along its world line until it is
half the round trip time before the reflection event, which will
cause a blue grid to appear. The two tick marks are now
simultaneous, so the clocks are synchronized, and serve as the
basis of a spacetime coordinate grid.</li>
<li>Drag the dot marker at the top of the left clock world line to
change the slope of the two clock world lines, so they represent
two clocks moving at a common speed. The gray world lines of
light flashes do not change slope, even if their source at the
left clock is moving. Repeat the synchronization exercise (1),
noticing that the slope of lines representing simultaneous events
has changed along with the clock speed.</li>
<li>Drag the dot marker at the top of the right clock world line
to change the spatial separation between the two clocks. Notice
that the lines of simultaneity, light the clock world lines, do
not change slope. Changing the scale of the coordinate grid
does not change which events are simultaneous.</li>
<li>Drag the dot marker at the top right of the gray diagonal
grid to change the slope corresponding the the speed of light.
This is a purely cosmetic change to show that our procedure for
constructing spacetime coordinates does not depend on how we
plot the diagram or on our choice of units for measuring time
or space.</li>
</ol>
<p>The surprise about this synchronization procedure, which amounts
to nothing more than correcting for the travel time of the
synchronizing light flashes, is that whether we judge two events
to be simultaneous depends on our state of motion: After
synchronizing the two clocks in the figure to reveal the line of
events simultaneous with zero time, try changing the slope of the
left world line and watch how the line of simultaneity also
changes slope. You will find that the phrase "at the same time as"
has no absolute meaning for distinct events; simultaneity
is <em>relative</em> to your state of motion. Two distinct events
which you judge simultaneous are not simultaneous from the point
of view of someone who is moving relative to you.</p>
<p>Furthermore, notice that as you explore all clock world line
slopes corresponding to velocities less than the speed of light,
the line of simultaneity sweeps over all the slopes corresponding
to velocities greater than the speed of light. That is, any line
closer to horizontal than the gray light-diagonals is the line of
simultaneity for some pair of clocks whose world lines are closer
to the vertical than the gray diagonals. We can therefore divide
directions in spacetime into two categories: Spacelike lines are
closer to horizontal than the gray grid lines, while timelike
lines are closer to vertical. These designations reflect the fact
that timelike lines are world lines for objects moving slower than
the speed of light, while spacelike lines are lines of
simultaneity for some observer moving slower than light. Every
straight line in the spacetime plane is either spacelike or
timelike, except for lines parallel to left and right moving light
flashes which mark the transition from spacelike to timelike. The
Euclidean plane has no analog for this dichotomy between timelike
and spacelike directions.</p>
<p>The operation of synchronizing clocks defines a unique spacelike
direction for any given timelike direction and vice versa. This
association between two directions in spacetime is the analog of
perpendicularity in the Euclidean plane. Euclid provides methods
for constructing perpendiculars to any given line in ordinary
plane geometry; synchronizing clocks using light flashes is a
method for constructing lines of simultaneity for any given world
line in spacetime geometry.</p>
<p>There is no place in this picture for observers moving faster
than light. Every direction in the spacetime plane is either the
direction of time or the direction of position for some
slower-than-light observer (or the direction of a light flash).
In fact, our clock synchronization procedure would not work for a
faster-than-light observer, because a light flash from their
trailing clock would never reach their leading clock. We will
explore later what happens when an observer accelerates forever,
but it will turn out that their world line never becomes
spacelike. There is no such thing as a faster-than-light
observer.</p>
<h3>Historical objections</h3>
<p>Every physicist before Einstein would have rejected our procedure
for synchronizing clocks.</p>
<p>Those like Newton who believed light consists of particles would
have rejected the idea that a light flash emitted from a moving
source would follow the same trajectory through empty space as a
flash emitted from a stationary source. They would insist the
light particles inherit the velocity of their emitter. From this
point of view, our synchronization procedure fails because we
assume a light flash from a moving clock will propagate along the
same line through spacetime as a flash from a stationary
clock.</p>
<p>Those like Maxwell who believed light is an electromagnetic wave
would have rejected our correction of the reflection time of the
synchronizing flash by half the round trip time. They would
insist that space is not empty but filled with ether, and that our
correction only applies when the clocks are at rest relative to
that ether. When this ether flows past the clocks, they would
insist that light travels faster downstream than upstream and the
proper correction for light travel time must take this directional
difference into account.</p>
<p>Ultimately, only very careful experiments with very accurate
clocks moving at very high speeds can answer these objections,
which come from the most authoritative sources possible. There is
no ether because we find that light always travels the same speed
through empty space no matter how fast you are moving, and light
emitted from moving sources travels at the same speed as from
stationary ones. More subtly, complicated theories (for example
that a moving light source drags some ether along with it) are
unnecessary, because the simple clock synchronization procedure we
have described here perfectly explains all the experimental
results.</p>
<p>Even the most surprising consequences of the fact that light
travels through empty space at the same speed relative to any
observer regardless of their motion were experimentally confirmed
in every detail within a few decades of Einstein's 1905
publication of his theory of special relativity. Some
consequences seem strange because we can have no direct experience
with things moving anywhere near as fast as a flash of light; to
our senses, light is present or absent instantaneously. In the
previous figure, you can drag the gray lines until both diagonal
directions collapse into horizontal lines. That is the limit in
which all our intuitions about motion were formed: Instead of a
whole range of distinct spacelike directions, there is only a
single spacelike direction, a line of constant absolute time.
Only when you are forced to work in situations where the speed of
light is not effectively infinite can you understand the
inadequacy of the concept of absolute time.</p>
<h2>Proper time and distance</h2>
<p>Simultaneity, defined by clock synchronization, determines the
direction of the spatial axis of a spacetime coordinate system,
given the direction of its time axis. In Euclidean geometry, this
is analogous to perpendicularity determining the direction of the
second axis of a rectangular coordinate system given the direction
of the first axis. But direction alone does not specify a
coordinate rotation - we also need to work out how rotated and
unrotated coordinates must be related in order to preserve
distance in the rotated coordinate system. In the Euclidean
plane, the Pythagorean theorem provides this relationship. We
must discover the analog of the Pythagorean theorem in the
spacetime plane.</p>
<p>We begin with the spacetime parallelogram which appears in our
clock synchronization procedure. The diagonals of this
parallelogram are world lines of oppositely directed light
flashes. Any such parallelogram is the spacetime analog of a
square in Euclidean geometry. Now consider another spacetime
square with a timelike edge in a different direction,
corresponding to a moving observer. How big does this new square
need to be in order for the moving clocks to measure a timelike
edge of the same duration as the stationary clocks measure for the
timelike edge of the first square?</p>
<p>Our clock synchronization procedure defines a unique spacelike
direction associated with every timelike direction, namely the
lines of simultaneity associated with a timelike world line. This
association is the spacetime analog of perpendicularity: A
spacelike line is "perpendicular" to a timelike line when the
spacelike line is a line of simultaneity for an observer whose
world line is the timelike line. The analogy in the Euclidean
plane is that two lines are perpendicular if aligning the first
line with the horizontal direction of the graph paper causes the
second line to be parallel to the vertical direction of the graph
paper.</p>
<p>Newton divided the spacetime plane into two regions, past and
future. For him, the present had no finite extent - it was just a
single line of simultaneity separating the half-plane of the past
from the half-plane of the future. In the Minkowski spacetime of
Einstein's theory of relativity, our practical procedure for
synchronizing clocks shows that the set of all spacelike
directions fills a finite area of spacetime, just like the past
future. The present has become a third region of spacetime. You
can watch the present collapse into a single Newtonian line by
changing the slope of the gray grid in the figure to be nearly
horizontal, corresponding to a nearly infinite speed of light.
Newton's conception of spacetime is very nearly correct when the
adjustment to account for the finite speed of light in our
synchronization procedure is negligibly small. Errors appear only
when material objects are moving at an appreciable fraction of the
speed of light, or when you need exceedingly accurate results.
Newtonian mechanics is not so much wrong as an approximation that
is usually very accurate.</p>
<p>The spacetime analog of a
Euclidean square is a parallelogram whose diagonals are parallel
to light flash world lines, like the bold parallelogram which
appears when you have synchronized the two clocks. Tiling
spacetime with these "square" parallelograms produces a coordinate
system in the spacetime plane analogous to square-grid graph paper
in the Euclidean plane.</p>
</div>
<script>
</script>
</body>
</html>