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| Oh demi tuhan, Norman! kamu seolah tidak pernah saja melihat sebuah lobang di kontinum ruang waktu sebelumnya. |
Tinjauan 3 Sifat Ruang :
Geometri, Topologi dan
Dimensi
Geometri Euclidean dan Geometri Non-Euclidean
Dalam wilayah penjelajahan akal budi manusia, telah lama timbul keyakinan yang sangat kuat dalam kemampuan manusia untuk memahami aspek tertentu kebenaran sejati alam semesta. Manusia berpikir jika kita mungkin berhasil dalam satu area penyelidikan ilmu maka kita bisa juga sukses untuk bidang yang lain. Sumber rasa percaya diri ini adalah kajian yang telah sangat tua dari geometri di mana Euclid dan orang Yunani kuno telah meletakkan landasan kokoh bagi disiplin ini.
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Geometri Euclidean adalah sebuah geometri di mana teorema Pythagoras untuk segitiga berlaku. Teorema tersebut memberikan jarak kuadrat antara dua titik (c2 pada diagram) sebagai jumlahan kuadrat dua sisi lain (a2 dan b2). Sebarang ruang pada mana jarak Euclidean ini berlaku disebut sebagai ruang yang datar (spatially
flat). |
| Fungsi jarak dapat diperumum ke sebarang jumlah dimensi ruang dengan menggunakan pola tepat sama sebagaimana di atas. Dalam kasus 3-D tinggal ditambahkan satu variabel bebas lagi saja. Diagram di sebelah kanan simbol h mewakili variabel baru ini. Variabel h tersebut adalah apa yang biasa diacu sebagai tinggi. |
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Geometri Euclid telah berjasa besar lebih dari sekedar membantu arsitek dan ahli pembuat peta (cartographers). Geometri ini telah membantu dikembangkannya suatu model penalaran, di mana kebenaran dideduksi dengan menerapkan aturan tertentu penalaran dari sehimpunan aksioma yang dianggap jelas dengan sendirinya. Teologi dan Filsafat mencontoh "metode aksiomatik" ini, dan sebagian besar argumentasi filsafat mengikuti pola umumnya. Dalam kasus ekstrim, sebagaimana ditampilkan dalam karya filosof Belanda Spinoza, proposisi filsafat bahkan diletakkan dalam pola definisi, aksioma, teorema, dan bukti sebagaimana dalam karya Euclid.
Geometri Euclidean dipercaya sebagai sebuah deskripsi tentang dunia, ia dipercaya bukan sekedar pendekatan, bukan juga sekedar konstruk manusiawi; namun pada mulanya geometri ini dipercaya sebagai sebuah kebenaran absolut. Para Teolog menunjukkan bahwa geometri ini adalah sebuah bukti bahwa pencaharian akan kebenaran absolut itu adalah mungkin dan masuk akal.
Di akhir tahun 1700an Immanuel
Kant , Filosof agung Jerman menyatakan bahwa geometri Euclid adalah benar terlepas dari pengalaman manusiawi kita.
Dalam paruh pertama tahun 1800an segala sesuatu mulai berubah.
Keyakinan kepada geometri Euclid mulai dicurigai. Matematikawan terkenal Carl
Gauss menunjukkan bahwa geometri tersebut sebenarnya telah ditentukan oleh eksperimen (eksperimen ...&; aksioma). Ada kemungkinan geometri lain yang menggambarkan kaitan erat antara titik-titik dan garis pada sebuah permukaan lengkung. Geometri ini juga dapat memiliki terapan praktis. Jarak yang jauh dipermukaan bumi membutuhkan geometri non-Euclidean.
Mengapa geometri non-Euclidean begitu tidak familiar bagi kita dari dulu hingga sekarang? alasannya karena manusia hidup sangat terbatas pada bagian tertentu permukaan bumi dan biasanya kelengkungan bumi sangatlah kecil untuk wilayah ini. Tukang bangunan dan kayu tentulah menggunakan geometri Euclidean namun pelaut yang mengarungi lautan luas tidak.
| GLOBAL:Di skala luas, kelengkungan struktur sangatlah jelas. |
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LOKAL: Di skala kecil kelengkungan struktur tidak mudah kelihatan. |
Kelengkungan ruang dapat dikelompokkan atas tiga kategori: Speris (kelengkungan positif), datar (kelengkungan nol), dan hiperbolik (kelengkungan negatif).
Dalam kelengkungan speris sudut di dalam segitiga jika dijumlahkan besarnya lebih dari 180 derajat.
Pada ruang datar, jumlah sudut-sudut sebuah segitiga tepat 180 derajat.
Dalam ruang Hiperbolik jumlah sudut segitiga kurang dari 180 derajat.
Ide ini disimpulkan oleh diagram berikut.
Dalam geometri Euclid semua permukaan adalah datar dan garis sejajar selalu berjarak sama, tidak pernah bersinggungan maupun berpotongan. Sebaliknya dalam geometri non-Euclidean, garis yang pada mulanya sejajar pada akhirnya akan saling bersilangan satu sama lain dalam kasus ruang lengkung berkelengkungan positif. Hal ini di ilustrasikan di bawah oleh dua ekor semut yang berjalan di sepanjang 3 tipe dari ruang tersebut.
Geometri ruang begitu penting bagi kajian kosmologi semenjak Teori Relativitas Umum Einstein, yang akan didiskusikan nanti, yang mendasarkan dirinya pada ide bahwa geometri ruang disebarang lokasi alam semesta secara langsung terkait dengan kuat medan gravitasi di lokasi tersebut. Makin kuat medan gravitasi, maka makin kuat pula kelengkungan ruang di situ. Dalam konteks kosmologis, tiga tipe kelengkungan tersebut adalah:
Alam semesta dengan kelengkungan positif adalah alam semesta yang akan mengembang hingga mencapai tingkat pemisahan tertentu antar galaksi lalu berkontraksi kembali ke ukuran nol. Alam semesta seperti ini disebut alam semesta tertutup.
The zero curvature universe corresponds to a universe that
will expand forever, slowing down as it does so. This is called a spatially
flat universe.
The negative curvature universe corresponds to a universe
that will expand forever. This is called an open universe. For a Escher print
based on the concept of negative curvature click on the following small picture.
Topology
Topology is the branch of mathematics concerned with the
ramifications of continuity. Topologist emphasize the properties of shapes that
remain unchanged no matter how much the shapes are bent twisted or otherwise
manipulated.
Such transformations of ideally elastic objects are subject
only to the condition that, for surfaces, nearby points remain close together in
the transforming process. This condition effectively outlaws transformations
that involve cutting and gluing. For instance a doughnut and a coffee cup are
topologically equivalent. One can be transformed continuously into the other.
The hole in the doughnut will be preserved as the hole in the handle of the coffee
cup.
| Topological
Classification |
| Objects with holes
can be classified topologically as follows: |
| No holes |
Genus 0 |
| One hole |
Genus 1 |
| Two holes |
Genus 2 |
| Three holes |
Genus 3 |
| EXAMPLES |
 |
The above shapes are
topologically equivalent and are of Genus 0
|
|
The above shapes are
topologically equivalent and are of Genus 1
|
| A wormhole is a genus
1 topological
defect in space. |
 |
Can you classify the following
topological equivalents?
Sometimes objects may have the same
genus type but they have a twist that makes then different
topologically. Consider the genus-1 doughnut shape shown above and the
genus-1 Moebius
strip shown below. The Moebius
Strip has a twist which allows the ants in this print to
continuously travel along both sides of the strip.
|
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When this twist idea is taken to 3-D with the Moebius
Strip becoming an object the twisted object is called a Klein Bottle.
This bottle is shown below in an animation which shows how you have to
twist things to get the object.
|
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What is the cosmological significance of topology? The part
of the universe that we can see is no more than 15 billion light years in
radius. This observable part of the universe shows no sign of having a weird topology.
However, we may not be able to see the crucial parts of the universe that might
lead to interesting topological observations. The following diagram gives
an illustration of what observation of cosmological topology might be like.
The diagram
below shows how the space can be wrapped around in a cylindrical
fashion.
|
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When the flat space containing the galaxies is
wrapped into a cylindrical shape and then further into a toroid
(doughnut) shape the galaxies have more than one way to 'see' each
other. Eventually light from one galaxy reaches the other by both
the short route and the long route.
|
| More than one way to 'see' translates into the
situation where observers in galaxies observe multiple copies of each
other. This is a cosmological prediction based on topology
considerations. |
 |
| In 1916 Albert Einstein put forward his theory of
gravity called General
Relativity. In this theory Einstein assumes that the effects of
gravity can be described in terms of the curvature of space and time
together. This hybrid 4-D space, upon which Einstein formulated his
theory, is called spacetime.
There are three dimensions of space in the four-dimensional spacetime
combined with one dimension of time. In General Relativity Einstein
links the curvature of spacetime with the way matter and energy are
distributed in the universe. |
 |
One can summarize the coupled arrangement
between matter and spacetime curvature in Einstein's gravity theory by the following statements:
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Mass (the source of the gravitational field)
tells spacetime how to curve.
|
|
Spacetime tells matter (any massive
body besides the source mass)
how to move.
|
Before Einstein's radical proposal about spacetime,
space was thought of as an unchanging stage upon which all the motions
and interactions of matter were played out. Space was like a tabletop
upon which transactions occurred independently of the structure and layout
of the table. Einstein's ideas however implied that space was highly
changeable. It was like a flexible material that accommodated every
massive object by curving appropriately in the local vicinity of the
object, much like a stretched sheet of elastic material would accommodate
itself to a heavy ball placed on its center area.
Curved space is understandable by means of these
analogies, but what does curved
time mean? Curved time means that the rate of flow of time is
determined by the strength of the gravitational field where it is being
measured. Time passes more slowly in strong gravitational fields. Time
is stretched out.
Einstein arrived at the idea that curvature is
gravity by thinking deeply about local effects of gravity versus global
effects of gravity. He deduced that we can always find a small piece of
space where gravity is locally zero. However, globally, gravity is
anything but zero. This local-global contrast led Einstein naturally to
the idea that curvature and gravity were linked. Curvature has the
same local-global character as gravity.
| Consider
the following statements and think about what the similarity is. |
| Look
at any curved surface in a small enough region and you will see
no curvature. |
Get
into the right frame of reference, that of a free fall situation
and you will feel no gravity. |
In 1908 Einstein had what he later called 'the
happiest thought of his life'. He had heard of a house
painter falling from the roof of a house. Einstein thought about
what the painter had said about the fall after it had happened. The
painter had experienced no negative repercussions while
falling. He had said that he just felt weightless. Only when the painter had hit the ground did he experience anything
of a negative nature. In the falling state, in effect, there
was no gravitational field. This 'no gravity' state can be
demonstrated in a similar situation with the dropping of some
objects while you are falling. What you discover in this experiment
is that the dropped objects do not fall. They stay right with you,
as if no gravitational field existed at all. In a freely falling
state (or alternatively, and equivalently, in a freely floating
state) gravity is abolished.
|
 Click
on the skydiving ('free float') picture to make it bigger.
What is the safest way to free float? The safest way is to
get yourself into orbit around the Earth. Then you are moving forward with a
certain speed while you are directly falling in towards the Earth. The moving
forward and the moving inwards compensate each other such that you continuously
encircle the curved Earth surface. You are continuously falling but you never
hit the ground.
In any free-float situation, such as that in a freely falling
spaceship, the paths of objects will never bend in any direction when they are
given a certain speed. These objects will move in completely straight lines.
|
Any object - a coin, a key, or a ball - follows a
straight track in a free-float spaceship.
|
As soon as the spaceship lands, and is not in a freely
falling frame anymore, a big difference occurs. Throwing the objects away
from you now makes the objects travel in a curved path towards the floor.
|
When the Earth (or a rocket)
pushes on the spaceship, the tracks curve relative to the
spaceship.
|
Question: What is the most natural situation? Why
does the curved path suddenly appear when the spaceship lands on the
Earth? Is there some mysterious force that causes the curved path of the
released objects?
From the Greeks to Galileo to Newton, the best
thinkers of the day tried to describe the force that made objects move
in this way. Einstein brought a revolutionary point of view to the
solution of the problem. Einstein said that there was no gravity force curving
the path of the objects. Einstein also said that the composition of the
objects had nothing to do with the curves. The curved
path is the fault of the
floor that forces us and all other objects away from the natural state of motion, the state of
free float. The electromagnetic forces between the atoms in the floor
push upwards, stopping our downward natural motion. This upwards force
makes us feel heavy.
Let the room be cut away at the
moment that we (who are in the room) throw the ball. The ball will
pass through the same space as before. The same ball thrown in the same
direction with the same speed appears to move quite differently when it is not in free float (left),
and when it is (right). Yet in the two pictures, the ball arrives after
the same time at the same location in spacetime.
|
|
The ball appears to move in the familiar curved path
which we have come to view (since the time of Newton) as the effect of a
gravitational force directed towards the center of the Earth.
|
With the
platform severed from its attachment to the Earth, the small house is in
a free-fall situation. This time the ball will move in a straight line,
unaffected by any so-called 'gravity' force. |
Einstein
recognized that falling is an illusion. The illusion arises from looking at
motion from a reference frame (the room at the left) that is not in free
float. The thing that keeps us from the free-float condition is the
electromagnetic forces from the atoms in the floor pushing up on our feet.
We, who stand securely, see things the wrong way around because the
ground beneath our feet is all the time pushing us away from a natural
state. This natural state is a state of motion equivalent to free
falling or free floating.
| To
be in tune with the universe we should be freely surfing the
spacetime curvature just as the Marvel Comics superhero Silver
Surfer freely surfed through his space. |
 click
to make bigger |
|
| The perverse way that
Einstein wants us to view gravity is summed up by the following
poem written by a 15 year old girl (Frances Ruml, granddaughter of the
gravity expert John A. Wheeler). |
What’s the fault of the
force on my feet?
What pushes my feet down on
the floor?
Says Newton, the fault’s at
the Earth’s core.
Einstein says, the fault’s
with the floor;
Remove that and gravity’s
beat.
|
What first strikes
us about the concept of free float is its paradoxical character. As a first
step to explaining gravity, Einstein got rid of gravity!
The second
feature that is evident to us is the local character of the free float
frame. Free falling rooms at different locations above the surface of the
Earth can be 'falling' in completely different directions. Free fall here
and free fall there are not the same. The concept only applies within a
limited local region. A way of measuring spacetime curvature is to look at
the difference between nearby free-fall frames. The bigger the difference ,
the bigger will be the gravitational curvature. If we lay down postage
stamps on a large spherical object, sooner or later we will notice
discrepancies in the covering of the curved surface. Similarly if we jump
into free fall frames all around a massive object, sooner or later we will
notice discrepancies in the motion of objects in nearby frames.
An example of curvature becoming evident concerns the use
of grids to mark out parcels of land on the Earth's surface by geographers.
Eventually the grids will not be adjacent to each other and the grid system
will not be withheld. See the picture below of the grid areas
(ten-mile-square townships) laid down in the flat prairie land of North
Dakota. Click the picture to make it larger. Even though the grid areas
start off nicely lined up, it does not take long for the curved surface of
the Earth to mess up the alignment.
Principle of Equivalence
The idea that a free fall frame is possible for any
object leads to one of the strongest predictions of Einstein's theory. If
all objects feel no force when they achieve a free float state then the
composition of materials will never be important for gravitation. If
something is in free fall, and it has no weight, then it can do nothing else
but float. A zero weight of gold is just
the same as a zero weight of feathers. They both should float weightless at
exactly the same height in a free fall frame. The principle that says that
all objects, no matter what their compositions, fall or float in exactly the
same way is called the Principle of
Equivalence. This
principle also has another form that equates acceleration with gravity. It
is not possible to distinguish, in a small region of spacetime, the
difference between the acceleration of an object and the existence of traditionally
postulated gravitational force.
It is easy to visualize curvature of space caused by
massive bodies. One thinks of objects sunken into the space.
However it is not so easy to think of curvature of the
hybrid space call spacetime. What does curvature of spacetime look like? To
get an idea of what this means we consider a simple experiment where two
balls are thrown across a room. Click here
to see the details of the experiment.
In Einstein's theory of
gravity, his famous equation, connecting Energy (E) with mass (m),
still holds. The speed of light squared is represented by the symbol
c squared in this equation. This c-squared term is a super-large
factor that gives even tiny amounts of mass equivalent to huge
quantities of energy
|
 |
| Since this equation holds we can deduce
that light, which has no mass, nevertheless must be affected by
gravity since it definitely has energy. |
Energy, by the above formula, must act
like a mass. Hence spacetime should tell lightrays how to move. This
is the crucial deduction that leads to one of the classic tests of
general relativity having to do with the bending of light by stars. |
The bending of light is seen everyday by astronomers. It
is possible for intervening massive galaxies to bend the light of even more
far-off galaxies in such a way that multiple images of the far-off galaxies
are produced. This effect is called a gravitational lens.
| The collection of galaxies in the
middle of the picture represents such a bending of spacetime that images
of galaxies behind the collection are multiplied. These multiple images
of the same extremely distant galaxy are shown encircling the central
mass collection. |
 |
General Relativistic
Predictions
BLACK HOLES
Black
holes form when matter accumulates in a small region of space. This can
happen at the center of galaxies such as our own galaxy the Milky Way
(which has a central 3 million solar mass black hole), or it can happen
when a huge star dies and does not throw off most of its mass when it
collapses near the end of its existence. This latter type of black hole
requires that the star have at least three times the mass of the Sun
before it will have sufficient gravity to completely trap light.
|
 |
One of the weirdest predictions of
General Relativity involves the idea of super-collapsed gigantic stars
that collapse so much that a tear in the spacetime fabric is created.
Light and nothing else is able to escape from these gravitationally
supreme objects. The puncture in the spacetime continuum is called a singularity. |
| Black Holes have an
interior region that cannot be observed. The boundary between where
light can potentially escape and where light and nothing else, can escape
is called the Event Horizon. The fact that
a singularity exists at the center of a black hole implies that the
Einstein equations are incomplete there. The radius of the event horizon
is called the Schwarzschild Radius. |
|
| Since the collapsed
stars that they are formed from are rotating, most black holes should
also be rotating. Rotating black holes are called Kerr
Black Holes. The picture to the left shows such a rotating black
hole with a jet of particles coming out of its central region. Astrophysical
objects have been observed with such high energy jets. No definitive
proof exists however that black holes exist. They are always inferred
from indirect evidence such as the radiation released when charged
particles fall into the black hole. Just because a jet comes out
from the center region of the black hole doesn't mean that the black
hole is emitting these particles. The jet phenomenon arises from
too many particles trying to get into the relatively small event horizon
of the black hole. So many things try to get in that some are
energetically flung out from the center by the intense pressure close to
the mouth of the black hole. |
|
Some people speculated in the past that the black
hole might be a way of traveling to another universe or to another part
of this universe. However, it can be shown that black holes pinch off as
soon as you continue down their throats. All matter that passes inside a
black hole gets crushed into nonexistence.
|
 |
 |
The violent deaths of large stars (supernovas) and the
collisions of extremely dense stars such as neutron stars with each
other can cause spacetime disturbances to happen and to spread out in a
wavelike manner. These waves are called gravity waves or gravitational
radiation. Within the next five years gravitational radiation should be
seen for the first time. |
| Below is an example of a supernova induced spacetime
wave disturbance. This disturbance can be detected by placing two masses
side by side and observing any change in their relative positions when a
gravitational wave passes by. Does the distance between them stretch or
compress? The answer has to be very high resolution since the effect is
so very small. Lasers are used to optically deduce the minute distances
that the gravity waves make the two masses travel. |
 |
| Recently there has been serious
scientific discussion of the idea of warp drive to boost spaceships to
apparent speeds considerably in excess of the speed of light. No known
way of actually doing this exists at the moment, but theoretically it's
not too hard to see how it might work. |
 |
| As the diagram shows, what one needs
to do is to compress the spacetime in front of the ship while expanding
the spacetime in the rear of the ship. The overall effect of doing this
would be to warp space in front of the ship so much that the distance
between where the sip is now and where it wants to be is lessened. Then,
although locally the ship is always traveling less than the speed of
light, the global transport is in effect much faster than the speed of
light. The captain of such a spaceship would have to keep well away from
other ships and communities while the ship was in warp drive mode since
much damage could be done to anyone nearby. |
The idea that global situations are quite different
from local situations runs deeply through everything in cosmology and
especially in general relativity. You have to be very careful when making
statements about the universe. Humans live only in an insignificant piece
of the overall universe. We have also only lived in this universe for a
negligible amount of time. Yet we can make deductions about regions that
we can never hope to reach and times that we can never hope to find out
about in a local way.
As an example of a local versus global tension, much
like that running through general relativity, consider the drawing by M.
Escher, a famous 20th century Dutch artist. Below we see a piece of that
drawing. Although it is somewhat otherworldly, the environment portrayed
seems to follow rules similar to what we are familiar with. This world is
however far stranger than what is shown.
Click on this picture to see
what the global situation is like.
| The Einstein Gravitational Field Equation
is shown below. This equation states that the spacetime curvature of
anyplace in the universe (left-hand side of the equation) must be equal
to the distribution of matter and energy in that part of the universe
(right-hand side of the equation). |
 |
It is this equation that was 50 years ahead of its
time when Einstein put forward his general theory of relativity in 1916.
|
We think of motion as being transitory. We think that being
stationary is normal. Yet in the overall scheme of things it is just the
opposite. It is impossible to be still in any absolute
sense. We can only be still relative
to other people or other objects in our vicinity. Even then every atom and
elementary particle in our body is engaged in motion.
Did you know that as you read this you are traveling at least
1000 kilometers per hour?
Below is a summary of the speeds that we are going right now.
Note that, although we are going at very fast constant speeds, we feel nothing.
Einstein put forward the idea that there were no
absolute speeds in the universe. He stated that whether we are moving at
a constant speed relative to something else, or that the something else
is moving at a constant speed relative to us, is not relevant. Either
point of view must be acceptable.
Being in a state of constant speed
causes no forces to be acting on you. Hence you feel nothing even though
you are zipping through the Milky Way at 900,000 km/hr. You only feel
something when you accelerate or decelerate to another speed. Just as
the free-float frames of reference were special in General Relativity,
the frames of reference where one travels at constant speeds are special
in Special Relativity.
| Einstein put forward
two
postulates that are encapsulated in what is called
The
Special Theory of Relativity. |
-
Light is special, in that all observers in
constant motion relative to somebody else will see exactly the same
speed for light in vacuum.
This postulate screws up space and time totally. If
everyone is to see light as going 1 billion km/hr then everyone has
to be measuring different space lengths and different time durations
to get the velocity of light to come out so that everyone gets the
the same 1 billion km/hr value.
-
Nobody can perform any experiment to show any
differences in physical phenomena when they are traveling at
constant speed relative to somebody else.
The laws of physics stay
exactly the same when the observer is in constant motion, no matter
how fast that motion is. If something blows up in a constantly
moving frame of reference then all constantly moving frames of
references will see the same thing blow up.
|
These two postulates essentially imply that whenever you move relative to someone else you enter into a distinct
temporal universe. You will measure time differently from the
other person and you will measure lengths differently from the other
person. Things, however, will conspire in such a way that everybody will measure
exactly the same value for the speed of light. Things will also conspire
to never allow causes to precede effects. Special relativity preserves
the causality of all situations.
|
The speed of light is a physically unattainable speed
for any object that has a mass. To accelerate a mass closer and
closer to the speed of light you need more and more energy. This can be
interpreted as the mass of the object getting larger and larger with
speed.
Two distinct events are said to be simultaneous if
they occur at the same time. If observers exist in distinct temporal
universes they will disagree about simultaneity. If they disagree about
simultaneity, then since you must measure the length of any moving
object by using the idea of simultaneity, lengths will not be the same
in the different temporal universe.
Why then do we not notice these funny things every
time we drive a car or ride a bus? The answer is that these
weird relativistic phenomena only happen when movements are close to one
billion kilometers per hour. No humans have ever gone this fast.
However little particles moving through the upper atmosphere do move
this fast. Particles being accelerated around rings such as those at the
large particle accelerator labs (Fermilab , CERN, etc) do move at these
speeds. Jets of charged particles coming from black holes and other
exotic astrophysical phenomena do travel at these speeds.
EXAMPLES
Galileo's Perspective
|
| The animated picture
below shows a train moving past an observer who is stationary relative
to us. There is a person on the train throwing a ball at constant speed.
The person standing sees the ball move at a constant speed equal to the
train's speed plus the speed that the person on the train actually
throws it. |
 |
Speed of Ball (according
to person on the ground)
EQUALS
Speed of Train + Ball's Release Speed (according to the
train person)
This is the way that people have been figuring out combined speeds
since the days of Galileo and Newton. In this type of world the two frames
of reference are within the same temporal universe. Only one universe time
is necessary. Light is expected to act exactly like the ball acts. |
| Einstein's
Perspective |
| The animated picture
below shows a train moving past an observer who is stationary relative
to us. There is a person on the train shining a light beam, which
necessarily goes at constant speed. The person standing does
not see the light beam move at a constant speed equal to the
train's speed plus the speed that the person on the train actually sees
it leave his flashlight. |
 |
Speed of Light
(according to person on the ground)
EQUALS
Speed of Light (according to the train person)
Light is special. This forces the train observer to live in a
different temporal universe from the ground observer. His time values
will be different for all events. |
Length Contraction
In order to measure the length of an object moving
relative to you it is necessary for you to get the two endpoints of the
object measured at exactly the same time. If you are sloppy and measure the
front endpoint of the moving object before the rear endpoint, then the rear
of the object will have moved ahead before you get it measured. You then
take the difference between the two values corresponding to the endpoints
you have sloppily measured and you will get the wrong length. It will be too
short because of your incompetence. Endpoints must be measured at exactly
the same time to get proper length measurements of moving objects.
However, nature is basically unfair in this respect. It
makes even the best observer look incompetent to any other moving observer.
Since each moving observer lives in a temporally distinct universe,
there will always be disagreements about the simultaneity of endpoint
measurements. One observer will always be measuring moving objects shorter
than what the other observer sees as their lengths.
The fact that any observer will always measure the
lengths of moving objects shorter that what their at-rest lengths are is
called the Length Contraction Effect. The
four pictures below illustrate this effect.
 |
The passing rocketship is going at 10% of the speed of
light. |
You have to go much faster than normal speeds, close to one billion kilometers
per hour, to see relativistic
effects.
 |
The passing rocketship is going at 87% of
the speed of light. |
Note that the vertical height of the spaceship is not affected by the
horizontal motion of the rocketship. That length stays the way it was when
the ship was stopped.
 |
The passing rocketship is going at 99% of the speed of
light. |
 |
The passing rocketship is going at 99.9% of the speed
of light. Nothing is left to see. |
Time Dilation
The different temporal universes that the theory of
Special Relativity demands for moving observers will have times such that
any moving observer's clock always ticks with dilated intervals. Hence a
moving clock will always seem to run slow. Your clock, in your rest frame,
will tick away, accumulating time much more rapidly than a clock in a fast
moving spaceship passing by. It is for this reason that people connect the
time dilation effect with future time travel. All you have to do is leave
home in a very fast spaceship. Travel for a certain distance and then return
to your home base. You will then be going into a future
| In the picture below
a rocketship reference frame travels at 99% of the speed of light
relative to some stay-at-home frame. The clock on the moving frame |
 |
| Less time is shown
on the moving clock since its tick intervals have grown 'fatter' as
it achieved its high speed. |
TWIN PARADOX
The
twin paradox arises from the time dilation effect. Say there are twin
brothers. One brother stays at home in a slow speed environment. The other
brother goes away in an ultra-fast spaceship. The "slow" twin ages
considerably (loses his hair). The younger twin returns still young (having
all of his hair). This situation is called the twin
paradox.
| In the example shown
below, the fast twin leaves the Earth at 75% of the speed of light. He
leaves as a baby. When he returns he has aged 37 years. The stay-at-home
twin has aged 57 years. |
 |
| Time dilation is a
relative effect. Each person sees the other person as moving. Hence each
person sees the other person's clock run slow. Each person is
legitimately allowed to claim that the other person's clock is the slow
clock. The reason why the fast twin gets younger is that he does
something that the stay-at-home doesn't do. To turn around, he has to slow down,
turn, and then
speed up again to get back to his home. It is this action, that the
stay-at-home twin doesn't experience, that forces
the time difference between the twins to be non-reciprocal. |
| Time dilation effects are happening
everyday in the Earth's atmosphere. Elementary particles that last no
longer than a few minutes in the laboratory have been tracked coming in
from outer space as cosmic rays. The surprising thing about the tracks
of these particles is that the normally short-lived particles are
definitely lasting for more than 20 minutes or more. We see them as
having their little personal clocks run slow. The particle however sees
its own clock running normally. In its frame of reference it lives no
longer than its laboratory lifetime. |
 |
The time dilation effect is taken into account every day to help keep the
atomic clocks on the 24 Global Positioning System (GPS) satellites that encircle
the Earth in sync with Earth-based atomic clocks.
|
 |
| Since the satellites are moving
relative to the atomic clocks that are on the ground they have dilated
time intervals. The Global Positioning System requires that the
satellite clocks be coordinated to a very high accuracy. This is
absolutely necessary to predict where the location is of people using
the GPS receivers on the Earth's surface. Even though the special
relativity effect is very very small, the atomic clocks are accurate
enough to be affected by the time dilation effect. |
|
WHAT IS TIME?
The fifth century Christian theologian Augustine
wrote: What then is time? I know well enough what it is, provided
that nobody asks me; but if I am asked what it is and try to explain, I am
baffled.
The passage of fifteen hundred years has, in fact, done
little to clarify the real meaning of time. Throughout history, human beings
have always puzzled over time's profound but seemingly vague nature. The
subject of time has fascinated poets, writers, and philosophers of every
generation. Modern science has given us a way of incorporating the concept
of time into measurements of physical quantities. We know that time is a
measure of change. We know that change involves the expenditure of energy.
By the second law of thermodynamics, energy expenditure produces unwanted
entropy. Thus time gets directly linked to the tendency of physical
systems to disorder. Time involves irreversibility.
| SATURN DEVOURING HIS CHILDREN (1824) by the Spanish
artist Francisco de Goya. |
 |
| Saturn was the roman name for Cronus, the god
associated in late Greek mythology with time. He swallowed his children
after being told that one of them would dethrone him. Only Zeus, who
later did dethrone him, escaped. The painting symbolizes how time ravages
and destroys all humans. Unlike Zeus, mere mortals cannot evade his
clutches. |
The irreversibility of time does not hold on small
scales. None of the major theories in physics seem to preferentially pick
out a certain direction in time. On small scales processes are genuinely
reversible. On large macroscopic scales, processes are unquestionably irreversible.
How this switch over, from irreversibility to reversibility, occurs is a
fundamental mystery.
The irreversible direction of time was not always
accepted by ancient civilizations. The cyclic character of time was a common
feature in Greek cosmological thought. Aristotle wrote that 'there is a
circle in all other things that have a natural movement and coming into
being and passing away. This is because all other things are discriminated
by time and end and begin as though conforming to a cycle; for even time
itself is thought to be a circle.' The Stoics believed that when the
planets returned to the same relative positions, as at the beginning of
time, the cosmos would be renewed again and again. The reason for the
acceptance of cyclic-time ideas was because time's arrow, a unique direction
of time, aroused deep fear, even terror, amongst the thoughtful of
ancient civilizations. A time direction implied instability, flux, and
change. It was associated with the end of the world and not associated with
rebirth and renewal.
It was the Judeo-Christian tradition that imposed a
'linear' irreversible time in Western culture. Through the Christian belief
in the birth and death of Christ and the Crucifixion as unique events, a
direction of time was accepted. These events were assumed to be
unrepeatable. Western civilization came to regard time as a linear path that
stretches between past and future. Before the Christians, only the Hebrews
and the Zoroastrian Persians had adopted this progressive view of time.
Scientists became involved with the concept of time when
Newton (1687) felt compelled to define time as being necessarily absolute.
This meant that all events could be regarded as having a distinct and
definite position in space and occur at a particular moment of time. This
moment of time was taken to be the same for observers everywhere in the
universe. Time was an unstretchable quantity, in terms of which, change in
the whole universe could be uniquely described. Newton constructed a deterministic
set of mathematical relations that allowed prediction of the future
behavior of moving objects and allowed deduction of the past
behavior of the moving objects. All that one needed in order to do this was
data in the present regarding these
moving objects.
Einstein changed these ideas totally. In his special
theory of relativity and his general relativity theory time became a
relative concept. This meant that time varied from place to place. Time was
stretchable.
|
In special
relativity time became stretched when you moved with any speed. The
effect
is biggest when you move at speeds close to one billion kilometers per
hour.
|
|
In general
relativity time became stretched when you placed yourself into
a
gravitational field. This effect is biggest when the gravitational field is
strong.
|
In relativity we can draw diagrams, called spacetime
diagrams, of the paths people take as they go forward in time. The path
is called the world-line of the individual.
| World-line of a person walking
to the right, stopping, and then walking back to the left. |
 |
| Note that the vertical
displacement of the individual is the time elapsed since she started
the journey. The individual cannot move in any way such that the
vertical direction is traversed in the negative (down) direction.
That direction would correspond to going backwards in time. |
On spacetime diagrams such as the one above, light rays
will travel along paths always at 45 degree angles to the horizontal and
vertical lines. Nothing can travel faster than these light paths. All
objects having mass must move slower that the light rays. Consequently the
motion of everyday objects falls within the 45 degree angled lines. These 45
degree lines form what is called the light cone.
Paths outside the light cone are forbidden. They represent object paths
corresponding to speeds greater that the speed of light.
| This is a diagram for the world-line
of a particle that travels at less than the speed of light. The place
where the two forbidden triangles meet is called the present. The future
is the triangular shaped region at the top of the diagram. The past is
the triangular shaped diagram at the bottom of the diagram. At no time
can the particle divert into the forbidden regions (red triangles)
without going faster than the speed of light. Every point along the
world-line of the particle must never be greater than 45 degrees. To go
backwards in time however this is exactly what the observer would have
to do. Hence, special relativity predicts that you can only go backwards
in time if you travel faster than the speed of light. |
 |
The effect of gravity on light cones is to tip them in
the direction of the gravitational curvature. Black holes tip the light
cones of nearby particles so much that the future part of the light cones
points directly into the black hole. In this case, no matter how the
particle moves, its future lies trapped inside the black hole.
Both the general and the special theory of relativity give
a method for time travel into the future.
In special relativity all one has to do is to leave the
Earth behind and travel in a super fast space ship. Your time intervals will
dilate, making your clocks run very slow compared with the clocks that remain
fixed to the Earth's surface. When you return to the Earth, you will have aged
a small fraction of the time that people who remained on the Earth have aged.
In effect you have traveled to the future of the Earth in your fast
'time-machine' spaceship.
In general relativity you leave the Earth and encircle a
dense object such as a neutron star for a small amount of time. To increase
the amount you travel into the future, you orbit closer to the neutron star.
The gravitational field dilates your time intervals. When you finally leave
the vicinity of the neutron star and return to the Earth, all inhabitants of
the Earth will have aged much more than you have. You have effectively
traveled to the future using a strong gravitational field to carry out the
time travel. For short mpeg movies simulating trips to neutron stars and black
holes click here.
The laws of physics do not preclude time travel. They seem
to work irrespective of the direction of time.
| There are, however,
two
logical objections to travel into the past in the same universe. They
are as follows: |
| 'Grandfather
Paradox'- You cannot travel to the past and disrupt some event in
such a way that you never existed in the first place. This is called the
grandfather paradox because you can think of going backwards in time and
killing your own grandfather. This is problematic, since if you really did kill
your own grandfather, you shouldn't exist at all.
|
'No Free Lunch'
Paradox'- Within the same universe time travelers
should not be able to bring information and energy to the past that could be used to create new ideas and
products. This would involve no creative energy on the part of the "inventor."
Example: Imagine that the famous artist, Pablo Picasso,
had traveled back in time to meet his younger self. He
could give his younger self his portfolio containing copies of his art. If
the young version of Picasso meticulously copied the reproductions, he could
have affected the future of 20th century art at a very early age. His reproductions
would exist because they were copied from the originals, and the originals
would exist because they were copied from the reproductions.
The time-traveling artist would have created masterpieces without ever
having expended any energy in their creation. |
It is possible to construct time machines in general relativity that
take you back into the past. These machines usually involve rapidly
rotating dense objects which are dragging spacetime around with them. An
example is the rotating massive cylinder shown below. You travel
forwards in time if you go around the massive rotating cylinder in its
direction of rotation. You travel backwards in time if you go around the
cylinder in the opposite direction that it is rotating.
Alternatively you can time travel using wormholes that connect our universe
with itself at a earlier time (or some other universe that is a copy of our
universe at a earlier time).
To avoid the two logical problems given above, it is
mandatory to allow for the existence of multiple worlds. It turns out that the
theory of physics, called quantum mechanics, that governs all physical
phenomena on the smallest scales can be interpreted literally as a theory of
parallel universes. Hence, quantum theory is quite consistent with the idea that
time travel can exist without logical difficulties.
How is time viewed in a parallel
universe scenario? Each moment of time is seen as a distinct
universe recreated from the previous universe such that the rules of existence
(the laws of thermodynamics) are obeyed. Below is a diagram of a stack
of 2-D universes. Time proceeds from the bottom of the stack to the top
of the stack. Time is then seen as just a universe label: Time 1 = Universe #1,
Time 2 = Universe #2, Time 3 = Universe #3, etc.
The recreation of the universe at any instant is done so
rapidly that no experiment can pick up any part of the universe's recreation at
that instant.
If you have two or more adjacent stacks of universes then it
is possible for time travel to be self-consistent. The time-traveler has to hop
from a certain time in the left stack of universes to a different time in the
right stack of copy universes.
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