I
want to take you on a journey. For me it started quite young, but
when at age twelve or thirteen my parents gave me a six-inch
Newtonian reflector, it began in earnest. I am told that my largesse
was the result of my sister getting (for a while) a pony, and their
was fear I would be envious, but the two events were never connected
in my mind and I never remember a trace of envy or resentment toward
my sister. I was just so darn happy to have the telescope. .
Even
in the rather light-polluted suburbia USA we lived in the telescope
revealed a marvel of heavenly capital the naked eye never suspected.
We all know there are craters on the moon, but with but 96X
magnification I could see them, bright and clear. And the Galilean
satellites of Jupiter. The rings of Saturn. And stars beyond stars,
nebulae, galaxies – by any professional standards it was just a
child’s toy, but what it brought into my backyard most of the
greatest philosophers of history could not have dreamt about.
It
took me … out there. Away from this secluded and narrow viewpoint
of tiny spot on planet Earth where I stood, toward places hundreds or
thousands of years of light speed travel that would have been needed
to actually be there in the flesh (not that I would have survived
long, but I never thought about that). It was a Asperger’s child’s
vision of paradise, to this day probably the best thing my parents
ever did for me. I was not to become an actual astronomer, but this
gift opened that door to me better than anything else.
I’d
like to repeat a figure from chapter one:
Figure I. (repeated)
Getting
away from our ordinary, Earth-bound existence and planting ourselves
somewhere in space – here, a position millions of miles beyond the
sun-Earth system – we see already how much our perspective on
things have changes. For one thing, the cause of the seasons, which
had baffled us before, becomes obvious. We also see Earth (and the
sun) as spherical objects in space, instead of as flat, infinite
surfaces which or may not have boundaries. If the picture were to be
fully fleshed out, we would see other planets too (specifically, the
inner world of Mercury and Venus, and the fourth planet Mars; whether
Jupiter and/or Saturn would show from here is not as clear).
Whether
you realize it or not, I have done something profound to your senses;
more precisely, to your brain’s interpretation of reality. But all
I have really done is change your point of view, as my telescope
changed mine. I call it profound, however, because the brilliant
insights of Albert Einstein, in the beginnings of the twentieth
century, demonstrated that it must be so.
* * *
If
you have ever taken high school physics, or a general college physics
course, you may have discovered that it is, in a very real sense,
boring. It’s boring because you’re not learning anything you
hadn’t already intuitively learned by about age two or so. If you
don’t believe that, then watch next time a magic show is performed
before a group of toddlers. They are just as dumb-founded and
thrilled seeing the laws of ordinary physics seemingly violated as
you or I. And even a small baby can tell when something is amiss; if
you convince it something is in a certain place and then reveal that
it isn’t, their eyes will open wide with surprise, and they may
even become distressed.
“But
that’s just common sense,” you might be tempted to protest. Yet
what is common sense? If it were as easy and as obvious as it seems,
artificial intelligence would be a snap to accomplish, and would have
been years ago.
The
fact is, our brains evolved to perceive and “understand” reality
in ways necessary for our stone-age ancestors and further back, and
thus it is unsurprising that we should possess, even at a very young
age, the common-sense concepts we collectively call reality. Recall
what I said about magicians and how they do what they do; they use
those “wired-in”, common-sense, laws of physics and manipulate
our senses and points of view to cause us to see impossible things.
Like bending a spoon with your fingers, if I may cite a rather common
trick by “psychics” – magicians who pawn themselves off as
special people with special powers.
What
you learn in basic physics course are the details, the precise
definitions, and the math behind the ordinary. And, despite what I
said, it isn’t boring at all; I strongly recommend taking such a
course (my mother did, and got a B, which somehow didn’t convey to
her that she could grasp scientific thinking), perhaps even before
reading what’s coming up.
* * *
The
lesson of the last section, I hope, is that although we possess
common-sense intuition about “ordinary” reality – the reality
all of us spend all our lives in, a reality within a narrow range of
space and time – at the same time evolution could not have bestowed
us no gifts about reality outside those strict ranges, because our
ancestors never encountered them. And indeed, it hasn’t. But
until you understand that, it is only natural that you should think
the Laws of Physics, as we somewhat pompously and arrogantly call
them, will apply everywhere, all the time, across all scales of time
and place.
That’s
why I started out talking about my childhood telescope, and showing
Figure I again. Even this is not too far a deviation from our
hum-drum down on this planet’s lives, but there are some noteworthy
differences. The biggest one may be that the light from the sun or
that reflected from Earth will take several minutes to reach our new
vantagepoint. We certainly aren’t used to significant (or any)
delays between the time something happens and when we observe it, for
light travels – well, it travels faster than anything known in the
universe, a full 186,282 miles per second. That being the case, when
a bank of lights at a stadium are turned on, the stadium is full alit
“at once” , though it actually takes around a fraction of a
millisecond or one thousandth of a second fir this miracle to happen.
Since our brains can’t measure time intervals that short (we’re
a tad slow, to tell the truth), this is instantaneously as far as we
are concerned. Indeed, by all common-sense measurements the speed of
light is for all practical purposes infinite.
Yet
it is in fact not
infinite, as our hovering over Figure I. shows. That light –
electromagnetic radiation I should call it, including radio,
microwaves, infrared (heat) rays, ultraviolet (black) light, x-rays,
and gamma rays – journeys at a specifically defined speed came out
of work on electricity and magnetism in the 1800’s, already dents
our common sense view of things. But what comes next tramples it
into unrecognizable shards.
I
have to backtrack some to explain why. I’ll ask you to close your
eyes (but don’t stop reading!) and imagine the following: someone
else and I are on a rail car, travelling at fifty mph down the track.
We both have baseball mitts (not actually necessary), and one
baseball, and we are playing a game of catch between us. Some kindly
passenger (you, as it turns out) on the train is timing how fast we
throw, and reports us both hurling the ball at 50 miles per hour, or
73 feet per second. You can easily calculate this because the rail
car is 73 feet long and it takes exactly one second from throw to
catch.
Open
your eyes again. That was probably easy to picture before your
mind’s eye, I’m certain; we do things like this all the time, if
never exactly this.
Okay,
close ‘em again, and this time picture yourself on a train platform
at a station, watching the train whizz by at fifty mph (it’s an
express, and doesn’t stop there). You can easily look through the
car’s windows and watch the game of throw and catch.
Question:
what do you see now?
You
are probably already uncertain as to whether you will see the same
thing, but if you haven’t quite figured out what you do see, I’ll
hand you the answer and then explain it. Using the same clock, you
now see the thrower at the rear of the car throwing, like a top
flight major league pitcher, the ball at 73 + 73 = 146 feet in the
one second that ticks off your clock, or one hundred mph; while the
thrower at the front can accomplish a mere 73 – 73 = 0 feet in that
second, or zero mph.
The
explanation is that speeds add. The train is travelling fifty mph
forward, and this speed must be added to the rear of car thrower
speed and subtracted from the front of car’s thrower’s velocity.
If all the window shades are drawn, however, the passengers have no
way of knowing their speed with
respect to the station,
because we can’t sense constant speed, only acceleration
(speeding up, slowing down, or changing direction). Everything here,
keep in your mind, is at constant speeds and directions. And I’ll
wager it doesn’t gall you too much. You’ve actually witnessed it
first hand many times in your life; you know you can’t really say
an object is travelling at such and such a speed without specifying
the reference point that speed is being measured from. Ever sat in a
motionless train and, while watching another train moving slowly by
you, actually sense yourself moving in the opposite direction? This
is why. Like magic. it confuses your brain again.
Let’s
get back to light. Nineteenth century physicists showed that visible
light was in fact an electromagnetic wave, or possibly some kind of
particle like a baseball only infinitely tinier, and the speed of
that wave was a well-measured 186,282 mps. The natural question now?
With
respect to what point of reference?
The answer is either: either none, because all the laws of physics
up to that point demonstrated that there are no privileged or special
points of reference in the universe, that they are all equal; or,
despite those laws (and, to their defense, a law of physics is so
only because we humans to our meager abilities have been able to
estimate it well enough to call it a law, so we can at least bend if
not outright break it when needed) their was something called the
universal aether,
a fluid having no density, no color, no resistance to movement
through it (viscosity), indeed no
observable properties at all,
which permeated all space and actually was stationary in some
absolute sense.
A
note of personal preference before I go on here. Lots of books on
relativity have lots of pictures of trains and trolleys and clocks
and other pertinent things – only natural, as this is what got
Einstein thinking about the issues raised here – but I won’t,
because if I haven’t presented my concepts in a simple,
straightforward enough manner, then I’ve already failed in the main
aim of this book. I assure you, it has nothing to do with my
incompetence in drawing pictures (OK, well it does some).
I’ve
given you the two possibilities and, if I have been successful, you
are probably up a creek paddleless trying to choose. How can there
be no point of reference for light’s speed; or, alternatively,
there is a point, the aether, that has no physical properties that
can be observed and measured? You should be reeling a bit by this
point, because by the late 19’th century almost all scientists were
reeling trying to answer this seemingly impossible conundrum. So you
are in good company.
I’ve
suggested that there are only two solutions to this problem, but, and
in fair warning this is where things get strange (but still
logical!), a third solution does present itself. Oh, how I wish I
could say I thought of it myself. No, it took the genius of an
Einstein to see what was so unobvious to the rest us. The third
solution is that light
is in fact it’s own reference point,
and all other speeds must pay homage to it. It is light,
electromagnetic radiation, to which we must bow to and follow it’s
rules, however absurd they may seem to us.
Again,
an aside before I move on. The speed of light is so enormous that,
compared to it, the differences between the ordinary speeds we
encounter (and the 50,000+ mph of space probes and objects, as fast
as they seem to us, are still way below light’s 670,000,000 mph, by
a factor of ten thousand and more) are so insignificant to render the
light speed problem moot. This is why we never notice it in our
lives, or ever imagine it in our minds. It is certainly not part of
our intuitive understanding of physics for it never had to be in our
evolutionary history.
* * *
I
apologize here, for I must lay down some equations for your
edification. They are not really complicated, not unless you’re
going to go into full-fledged physics mode, which there is no need to
do. The first equation involves a quantity that you’ve probably
heard of, momentum. It is simply the mass of an object multiplied by
its velocity (speed + direction, remember). Prior to Einstein, and
as still presented in all general physics courses today it is:
momentum(
or p) = mass(or m) × velocity(or v)
p
= mv
Equation I.
This
is the equation Newton derived for what, essentially, he called
inertia, and so derived the now famous Law of Conservation of Linear
Momentum (quite similar to the one for Angular Momentum we’ve
already encountered). Newton derived it assuming that all velocities
were relative to some point of reference, even light, although he had
little idea what light was or how/how fast it travelled. Science
wasn’t developed well enough in his day, through no fault of his.
In
Einstein’s new formulation, the equation of momentum must be
modified from p = mv
to:
p
= m0v
Equation II.
Where
or gamma (mathematics is all symbolized, even X + Y = Z, so don’t
let this throw you), stands for:
Equation
III.
The
naught (0) on m
indicates rest mass, which we’ll come to in a tick, and c
means the speed of light.
Again,
don’t let any of this get too heavy on you; you should have had all
the symbolisms in high school, or can reference them easily. Anyway,
this factoring in of the square root (what √¯
means) of 1
– (v/c)2
to the momentum equation, which remember, applies to all objects,
takes into account c
(speed of light) being this supreme reference point in the universe
we must all be subservient to. Note something critical here: if v
<<
c,
as is the case with all velocities we normally encounter, then (v/c)2
goes to essentially zero, and
just as essentially goes to 1, meaning that gamma is just 1/1 or 1
and drops out of the equation, leaving us with our Newtonian
original, except for that lingering naught on m
which we’ve let to explain beyond calling it the rest mass. Is
there such a thing as a different not at rest mass?
Yes
there is, and a little examination of Equation
III. should
show why. Imagine we make v
very close to c,
or even make it equal to c.
Then (v/c)2
becomes just 1, and as 1 - 1 = 0 the bottom half of
becomes infinite, meaning that p becomes
infinite too!
Even
as v
gets
closer and closer to c,
p grows
rapidly. It is as though all the energy we are throwing into our
object to make it go faster and faster end up only increasing its
mass as the speed of light is approached. If we could get to c
the mass would be infinite in fact.
The
only possible physical interpretation of this is that energy and mass
are somehow equivalent, and dumping more of the former onto an object
means it also has more of the latter. The math for that is just
hairy enough to excuse us from examining it but the bottom line is
the equation we all know and love: E =
m0c2.
The naught after m
is, as said, the object’s rest mass; multiplying it by the speed of
light squared gives its equivalent in energy, and since c2
is a very large number you will see (as scientists in the 1930’s
were beginning to see, the results being Hiroshima and Nagasaki), if
you can make the conversation you will release a very large amount of
energy indeed.
Other,
equally strange phenomena crop up when we move close to the light
speed. Because all observers must find the same value for the this
speed, regardless of their reference points, time and space become,
well, malleable; it can be different with different observers.
Let’s
go back to our baseball throwers in the rail car. The big
difference, instead of throwing baseballs back and forth and each
other, each one now has a laser pointer. As soon as the light from
one pointer reaches the corresponding “catcher”, he in turn
flashes his collaborator with his own pointer.
This
experiment, as described, may sound absurd. The laser light travels
so fast that it takes only 7.422×10-8
seconds, or 74 billionths of a second, for the light to exit the
pointer and reach the other player, still fifty feet away. Very well
then; let’s makes this easier to visualize by making the rail car
186,282 miles long, so that it now takes our full second for the
traverse time. Never mind that this makes the experiment physically
impossible (the rail car would stretch around our world over seven
times!). Experiments can, in many cases are, done in our minds; as
long as we get the math right and imagine things correctly, this is a
perfectly valid approach to the subject. (Such experiments are
called gedanken,
a German word meaning literally ”done in thought”).
Do
you have the picture in your head, though admittedly it is a bit
tougher this time? Good. Next: imagine yourself, as before, the
measurer in the rail car, timing how long it takes the laser light to
get from pointer to receiver, and back. Naturally you find this
number to be 186,282 mps, just the speed of light. No surprises
there.
Now
place yourself on the station again. You time the laser pulses
again, just as you timed the baseball throw before from the platform.
Now, one, crucial, last part of the experiment to stitch into your
mind’s eyes: the rail car is also travelling 186,282 mps pass the
station, in the same way as it was travelling 50 mph before. So:
what do you expect to see?
With
the baseballs, we obtained the answer that the speeds were either 50
+ 50 = 100 mph for the back of car thrower, or 50 – 50 = 0 mph for
the front of car, for now our reference point is the platform at the
train station. Speeds, velocities more precisely, are additive
because they depend on the point of reference you are measuring them
from.
Amazingly,
the result is completely different for the laser pointers and the car
travelling at the speed of light! Recall
my statement that the speed of light is its own reference. This
means that its speed is always the same, regardless of any other
point of reference. The platform observer, pocket watch in hand,
finds that the laser pointers still fire their light beams at each
other at c,
just as the car observer does. Indeed, every observer, wherever he
is in space and time, obtains the same values. There is no adding of
speeds, at least not the way it was with baseballs.
If
you are trying to make sense of this, and failing, I can tell you
why. You are making the common-sense assumption that space and time
are the same for all viewers. But that is incorrect. It feels
right only because all your experience comes from speeds much below
c,
and evolution by natural selection has been geared to that. But our,
rather simple, gedanken
experiment has shown that both time and space are malleable, and
depend on the observer’s state of motion.
Space
and time are not flat and absolute. Special relativity shows this
beyond a doubt. But if they are not flat and absolute, then – what
are they? It was to take Einstein ten more years to work that out,
and so much of modern cosmology depends on what he discovered. It is
time to turn to there, the next state: the general
theory of relativity, published in 1916.
* * *
Einstein
was a person gifted with deep imagination and insight into nature.
He knew that if he tried to work out his ideas about light and space
and time not only with static motion but with accelerated motion as
well, he would probably never have succeeded at either. So he
started with static, meaning straight line, constant speed, motion,
and in 1905 arrived at his special theory. It’s special because it
specifically excludes all motions which involve changes in speed or
direction (collectively known as accelerations), and concentrates
purely on static motion. It was a maneuver which rewarded him with
pure gold. It showed that mass and energy were equivalent, that the
speed of light was the ultimate speed in the universe, and perhaps
most importantly, that time and space were not flat, abstract
constructs, the same for everyone everywhere, but ebbed and flowed
depending on different circumstances. An enormous triumph, which
earned him little notice at the time (he was still working at the
same Swiss patent office well after publication), and which decidedly
did not impress those who bequeathed Nobel Awards (though he did earn
one some years later for a different line of work, which we’ll
discuss in the next chapter).
Getting
back to the theme of this section, special relativity was just that;
a special case of relativity at work. Einstein’s main goal was a
general
theory of relativity, one which included accelerated motion as well
as static. It was to take him another ten years to grind out the
final, so-called field
equations for relativity, equations I will not present here, fully
confessing I don’t know enough math to do so.
We
don’t need to understand that math (few do) to get a decent sense
of what general relativity is about. One of the things Einstein
noticed when dealing with accelerated motion was the results were
virtually indistinguishable from being in a gravitational field.
Here on Earth, as we would on all mass bodies in the universe, we
feel ourselves being pulled downward by some unknown, almost magical
force, which we call gravity. And indeed, if the ground beneath our
feet were to give way, we would fall, ever faster and faster, into
the resulting abyss, until we were smashed to our deaths by whatever
we finally landed on (if the opening goes deep enough, however, we
will be incinerated by the heat of Earth’s depths and crushed by
the overwhelming pressures first – this all assumes the fall is far
enough, of course).
An
aside to describe accelerated motion. Our falling would occur under
a force of one g
(the gravitational force at Earth’s surface), meaning that we would
plunge ever faster downward, at a rate of 32 feet per second per
second, or 32fps2.
All this means is that, after the first second, we are travelling at
32 fps, after the second second 64 fps, after the third 96fps, and so
on. (All this ignores air friction, which counteracts the
acceleration and limits us to 200-300 fps final speed, depending on
our orientation.) Acceleration means a change in velocity, faster or
slower (or in a different direction) over time.
All
sorts of phenomena call cause accelerated motion. Take, as I believe
Einstein did, an elevator. If you were sealed inside a windowless
elevator somewhere out in deep interstellar space, you would be
weightless and float freely all about the car. Now imagine that some
unknown being of prodigious powers attaches a rope to the top of the
elevator car, and starts pulling it upward with a force of one
g
(32fps2,
remember). I think you intuitively sense that you would be slammed
onto the floor of the car, and you would be right – though be
careful about intuitions! I ask you a question: now that you are
standing on a floor with a 32fps2
acceleration force trying to pull you downward, what is the
difference between this situation and being in an elevator on Earth,
at a stopped floor level? Remember there are no windows, nothing to
tell you what is going on outside the elevator except this force
pulling you down.
If
I tell you basically nothing, you are probably not surprised. After
all, we have all been in elevators, and felt the lurch in our
stomachs, and the temporary lightness or heaviness in our bodies as
we moved up or down in the car (try this at the Empire State Building
or another monster skyscraper with express elevators if you really
want to feel this effect). The sensation of falling if an elevator
plunges down rapidly can be quite unnerving, rather like being on a
roller coaster. I still have dreams about it, once in a while.
So
our super being accelerating us at 32fps2
“upwards” (there is no such thing as direction in deep space,
remember) feels exactly the same as being on an elevator on Earth,
stopped at a specific floor. The being could even lurch us “upward”
at more or less than one
g
if it really wanted to mimic the effects of the ground-based car.
The point being, you simply could not tell the difference.
Here’s
where an Einstein mind works on a different plane than our own. He
made the deduction that, the scenario described above being true,
there
was no difference, according to the laws of physics, between them.
They are in essence one and the same phenomena. Like me, I suspect
you’ll have to put this book down at this point and chew on this
idea. If there is no difference in the perception of two or more
experiences, than physics tells us they are intimately linked by some
thread; essentially, they are the same thing. This most certainly
does violate our intuitions, which accounts for the warning I gave
before.
* * *
But
what is the thread connecting the two? Einstein finally realized
that space and time, and their malleability, had to be brought into
the situation to make that thread. But another, hopefully short,
aside on another needed prerequisite is needed here to show how. You
have probably heard of the concept called entropy,
though you might (and should be) puzzled as to its exact meaning. We
won’t get into that here, except to say that one meaning is that
physical systems are inexorably (actually, probabilistically) drawn
to their lowest energy states. An example is the falling we’ve
been talking about; think about the energy, derived from its load of
jet fuel, it takes to keep an airliner miles above the ground, or the
far greater amount of rocket fuel it takes to put an astronaut on the
moon or a robot about the planet Saturn. Falling things lose energy,
which, according to the laws of entropy, is exactly what they should
and must do, unless something stops them, like the floor of an
elevator car.
The
elevator, whether on Earth or in deep space, is providing energy to
keep its occupants out of a state of free
fall,
i.e., what they’d be doing without the forces acting on them,
floating about in a zero gravity (g)
field. In both cases we are dealing with acceleration, even for an
elevator stationary on Earth’s surface. What could account for
such a curious conundrum?
Space
and time. Spacetime. Special relativity had indicated that the two
were indivisibly connected, and the fertile mind of the Swiss patent
clerk finally become modest physics professor worked instinctively
along these lines. If spacetime were a flexible, or curvable,
concept that changed as an observer’s reference point changed, then
maybe it was more that that: if was a real
thing,
and not only real but highly variable depending on perhaps many
conditions. One of those conditions, he realized, was the presence
of mass. He realized that mass objects somehow distorted spacetime,
in a way that caused other objects to be attracted toward them.
Using this insight, he finally solved the problem of the cause of
gravity, which Newton knew better than to destroy his reputation on
solving two hundred years earlier.
To
explain how spacetime operates under general relativity I could
reproduce a fairly standard picture, chances are you’ve seen, which
appears to illustrate it:
Figure
IV.
The grid represents
the “curvature” of spacetime about the massive body of the planet
Earth. Although it gives a good feeling for what is going on – you
can easily picture objects approaching Earth being drawn in by the
curvature of spacetime (the white grid), but the picture works only
because Earth is at the bottom of a deep well and we intuitively know
that objects near a well will be drawn into it. What I am objecting
to in this picture as that it assumes what it is supposed to be
explaining. It’s not a bad start but it must not be an end to the
explanation. The reader wants to know what is really happening.
If instead of
falling downwards the gridlines (representing spacetime, remember)
get closer together as they approach Earth, then I believe we have a
better picture of why objects fall toward other objects. Objects
cause spacetime to compress, rather than to indent; a very small
compression for objects the size of Earth, larger for the size of the
sun, and quite large for nasty things like neutron stars and black
holes, which compress them into singularities,
a place which we won’t cover here.
Recall entropy.
Objects “prefer” to be at the most compressed regions of
spacetime because they can lose energy that way and obtain entropy by
this particular meaning. That’s why they’re attracted toward
each other, and will either fall into each other or orbit each other,
the latter until loss of energy results in an eventual (trillions of
years and more for the Earth-sun system, so don’t worry) collision.
If
spacetime can curve in one way, it can – at least in theory –
curve many ways – and many of the solutions to Einstein’s field
equations describe the entire universe as being “closed”
(spherically even) or “open”, or “saddle-backed” or plain old
flat. New solutions crop up once every few years or decades,
describing how the universe must be or how it will evolve in time, or
what phenomena (like black holes) it might reveal. There is also a
“standard” solution, which itself gets itself upgraded once in a
while, as the result of new astronomical observations (e.g., that the
universe is not only expanding but expanding more and more so, due to
something called “dark energy”) and other considerations. .
* * *
I am going to stop
here, because if you have been following me reasonably well, then
I’ve succeeded in what I set out to do in this chapter. I wanted
to describe for you a field of physics which, although it violates
all our common-sense notions about how things behave and why, was
within your grasp to at least get a good feel for. Don’t worry
about all the sundry details I’ve left out (though if this has left
you hungering for a deeper understanding of relativity, all power to
you!); I just wanted to give you a taste for how strange and
wonderful reality can be when we drag ourselves away from our
evolutionary-derived intuitions. Again, if I’ve succeeded, than we
are ready to talk about what I’ve been setting forth to cover, some
chemistry and biology that should be within your understanding as
well. This is where we start on a very new road from the physics of
Einstein, to the physics (and chemistry) of the quantum, a field many
great minds have contributed to and to which, I think, no one can be
designated the creator of. On to quantum cats.
