# The Celestial Sphere

Humans perceive in Euclidean space -> straight lines and planes. But, when distances are not visible (i.e. very large) than the apparent shape that the mind draws is a sphere -> thus, we use a spherical coordinate system for mapping the sky with the additional advantage that we can project Earth reference points (i.e. North Pole, South Pole, equator) onto the sky. Note: the sky is not really a sphere!

From the Earth’s surface we envision a hemisphere and mark the compass points on the horizon. The circle that passes through the south point, north point and the point directly over head (zenith) is called the meridian.

This system allows one to indicate any position in the sky by two reference points, the time from the meridian and the angle from the horizon. Of course, since the Earth rotates, your coordinates will change after a few minutes.

The horizontal coordinate system (commonly referred to as the alt-az system) is the simplest coordinate system as it is based on the observer’s horizon. The celestial hemisphere viewed by an observer on the Earth is shown in the figure below. The great circle through the zenith Z and the north celestial pole P cuts the horizon NESYW at the north point (N) and the south point (S). The great circle WZE at right angles to the great circle NPZS cuts the horizon at the west point and the east point (E). The arcs ZN, ZW, ZY, etc, are known as verticals.

The two numbers which specify the position of a star, X, in this system are the azimuth, A, and the altitude, a. The altitude of X is the angle measured along the vertical circle through X from the horizon at Y to X. It is measured in degrees. An often-used alternative to altitude is the zenith distance, z, of X, indicated by ZX. Clearly, z = 90 – a. Azimuth may be defined in a number of ways. For the purposes of this course, azimuth will be defined as the angle between the vertical through the north point and the vertical through the star at X, measured eastwards from the north point along the horizon from 0 to 360°. This definition applies to observers in both the northern and the southern hemispheres.

It is often useful to know how high a star is above the horizon and in what direction it can be found – this is the main advantage of the alt-az system. The main disadvantage of the alt-az system is that it is a local coordinate system – i.e. two observers at different points on the Earth’s surface will measure different altitudes and azimuths for the same star at the same time. In addition, an observer will find that the star’s alt-az coordinates changes with time as the celestial sphere appears to rotate.

Celestial Sphere:

To determine the positions of stars and planets on the sky in an absolute sense, we project the Earth’s spherical surface onto the sky, called the celestial sphere.

The celestial sphere has a north and south celestial pole as well as a celestial equator which are projected reference points to the same positions on the Earth surface. Right Ascension and Declination serve as an absolute coordinate system fixed on the sky, rather than a relative system like the zenith/horizon system. Right Ascension is the equivalent of longitude, only measured in hours, minutes and seconds (since the Earth rotates in the same units). Declination is the equivalent of latitude measured in degrees from the celestial equator (0 to 90). Any point of the celestial (i.e. the position of a star or planet) can be referenced with a unique Right Ascension and Declination.

The celestial sphere has a north and south celestial pole as well as a celestial equator which are projected from reference points from the Earth surface. Since the Earth turns on its axis once every 24 hours, the stars trace arcs through the sky parallel to the celestial equator. The appearance of this motion will vary depending on where you are located on the Earth’s surface.

Note that the daily rotation of the Earth causes each star and planet to make a daily circular path around the north celestial pole referred to as the diurnal motion.

# Some Important Geometry. (Astronomy)

Great Circles:

The shortest path between two points on a plane is a straight line. On the surface of a sphere, however, there are no straight lines. The shortest path between two points on the surface of a sphere is given by the arc of the great circle passing through the two points. A great circle is defined to be the intersection with a sphere of a plane containing the center of the sphere.

Two great circles

If the plane does not contain the center of the sphere, its intersection with the sphere is known as a small circle. In more everyday language, if we take an apple, assume it is a sphere, and cut it in half, we slice through a great circle. If we make a mistake, miss the center and hence cut the apple into two unequal parts, we will have sliced through a small circle.

Two small circles

Spherical Triangles:

If we wish to connect three points on a plane using the shortest possible route, we would draw straight lines and hence create a triangle. For a sphere, the shortest distance between two points is a great circle. By analogy, if we wish to connect three points on the surface of a sphere using the shortest possible route, we would draw arcs of great circles and hence create a spherical triangle. To avoid ambiguities, a triangle drawn on the surface of a sphere is only a spherical triangle if it has all of the following properties:

• The three sides are all arcs of great circles.
• Any two sides are together greater than the third side.
• The sum of the three angles is greater than 180°.
• Each spherical angle is less than 180°.

Hence, in figure below, triangle PAB is not a spherical triangle (as the side AB is an arc of a small circle), but triangle PCD is a spherical triangle (as the side CD is an arc of a great circle). You can see that the above definition of a spherical triangle also rules out the “triangle” PCED as a spherical triangle, as the vertex angle P is greater than 180° and the sum of the sides PC and PD is less than CED.

The figure below shows a spherical triangle, formed by three intersecting great circles, with arcs of length (a,b,c) and vertex angles of (A,B,C).

Note that the angle between two sides of a spherical triangle is defined as the angle between the tangents to the two great circle arcs, as shown in the figure below for vertex angle B.

Earth’s Surface:

The rotation of the Earth on its axis presents us with an obvious means of defining a coordinate system for the surface of the Earth. The two points where the rotation axis meets the surface of the Earth are known as the north pole and the south pole and the great circle perpendicular to the rotation axis and lying half-way between the poles is known as the equator. Great circles which pass through the two poles are known as meridians and small circles which lie parallel to the equator are known as parallels or latitude lines.

The latitude of a point is the angular distance north or south of the equator, measured along the meridian passing through the point. A related term is the co-latitude, which is defined as the angular distance between a point and the closest pole as measured along the meridian passing through the point. In other words, co-latitude = 90° – latitude.

Distance on the Earth’s surface is usually measured in nautical miles, where one nautical mile is defined as the distance subtending an angle of one minute of arc at the Earth’s center. A speed of one nautical mile per hour is known as one knot and is the unit in which the speed of a boat or an aircraft is usually measured.

Horizon System:

Humans perceive in Euclidean space -> straight lines and planes. But, when distances are not visible (i.e. very large) than the apparent shape that the mind draws is a sphere -> thus, we use a spherical coordinate system for mapping the sky with the additional advantage that we can project Earth reference points (i.e. North Pole, South Pole, equator) onto the sky. Note: the sky is not really a sphere!

From the Earth’s surface we envision a hemisphere and mark the compass points on the horizon. The circle that passes through the south point, north point and the point directly over head (zenith) is called the meridian.

This system allows one to indicate any position in the sky by two reference points, the time from the meridian and the angle from the horizon. Of course, since the Earth rotates, your coordinates will change after a few minutes.

The horizontal coordinate system (commonly referred to as the alt-az system) is the simplest coordinate system as it is based on the observer’s horizon. The celestial hemisphere viewed by an observer on the Earth is shown in the figure below. The great circle through the zenith Z and the north celestial pole P cuts the horizon NESYW at the north point (N) and the south point (S). The great circle WZE at right angles to the great circle NPZS cuts the horizon at the west point and the east point (E). The arcs ZN, ZW, ZY, etc, are known as verticals.

The two numbers which specify the position of a star, X, in this system are the azimuth, A, and the altitude, a. The altitude of X is the angle measured along the vertical circle through X from the horizon at Y to X. It is measured in degrees. An often-used alternative to altitude is the zenith distance, z, of X, indicated by ZX. Clearly, z = 90 – a. Azimuth may be defined in a number of ways. For the purposes of this course, azimuth will be defined as the angle between the vertical through the north point and the vertical through the star at X, measured eastwards from the north point along the horizon from 0 to 360°. This definition applies to observers in both the northern and the southern hemispheres.

It is often useful to know how high a star is above the horizon and in what direction it can be found – this is the main advantage of the alt-az system. The main disadvantage of the alt-az system is that it is a local coordinate system – i.e. two observers at different points on the Earth’s surface will measure different altitudes and azimuths for the same star at the same time. In addition, an observer will find that the star’s alt-az coordinates changes with time as the celestial sphere appears to rotate. Despite these problems, most modern research telescopes use alt-az mounts, as shown in the figure above, owing to their lower cost and greater stability. This means that computer control systems which can transform alt-az coordinates to equatorial coordinates are required.

# Dualities (String Theory)

Sometimes different mathematical theories describe the same physics. We call this situation a duality. In many cases, calculations which are very complicated in one theory become much easier in the other.

Usefully, string theory is awash with dualities. These variously offer us new perspectives on reality, improve our ability to compute hard sums and unite disparate areas of physics. Much of modern research focuses on using these dualities to better understand a broad spectrum of topics.

T-duality is the simplest to appreciate. Remember that string theory requires six extra dimensions tightly curled up in space. Naively, one would think that these dimensions could be arbitrarily big or arbitrarily small, with different physics holding in each case. However something strange happens when you make these dimensions very small. Of paramount importance is a tiny quantity known as the Planck length, which we denote by a.

How does the radius of a circular dimension affect the physics of string theory? We can appreciate this work with a thought experiment.

Set up a circular extra dimension the size of the Planck length. Start contracting the circle and measure the resulting physics. Your readings will vary depending on the size of the dimension. Now repeat the experiment, but with a crucial difference; instead of contracting your circle, expand it.

Observing the physics again, you realise that it’s exactly the same as for a contracting dimension! There is a duality between the two scenarios. Mathematically it can be proven that extra dimensions with radii r and a/r produce the same physics: they are identical theories.

An extension of T-duality produces mirror symmetry. In many string theory models, the extra dimensions form a six dimensional shape called a Calabi-Yau manifold. Sadly there are millions of different Calabi-Yau surfaces, each with a slightly different structure. The properties of the Calabi-Yau manifolds affect the expected four-dimensional physics. So we must pin down the correct possibility for the physics we observe.

This requires a lot of calculation. And maths is hard in six dimensions, as you might guess! But here’s where mirror symmetry comes in. In the late 1980s it became clear that Calabi-Yau shapes come in pairs. For any given pair, both lead to the same physics. We have a duality! Physicists could chop and change between mirror pairs, making computations more tractable.

Our third duality is more fundamental: it underpins the success of M-theory. We’ll refer to it as S-duality. All quantum field theories contain a coupling constant, which determines the strength of interactions between particles. String theory is no exception. The value of the coupling constant vastly affects the behaviour predicted.

During the First Superstring Revolution physicists realised that there were five different brands of string theory. At first it seemed like they were all completely separate. But the discovery of various S-dualities sparked a paradigm shift. These dualities related the different flavours of string theory through a framework called M-theory.

More precisely physicists paired up the different types of string model, like so. Take two distinct string theories, A and B. They each have an adjustable coupling constant. If A has a large coupling constant and B a small one, then they predict exactly the same physics. The end result was that the many different string theories were united under a single banner.

Finally we come to the hottest guy in town. The AdS-CFT correspondence is a conjectured duality which has been around for barely a decade. Subtle yet powerful, it has profound implications for string theory as a tool in research. It’s such an important idea that it requires a full explanation.

# Einstiens General Relativity(FullyExplained)

 Spacetime and Energy
 relativity unifies space, time, mass and energy Special relativity and E=mc2 led to the most powerful unification of physical concepts since the time of Newton. The previously separate ideas of space, time, energy and mass were linked by special relativity, although without a clear understanding of how they were linked.

 explanation provided by general relativity, where a complete theory of gravity is provided by using the geometry of spacetime The how and why remained to the domain of what is called general relativity, a complete theory of gravity using the geometry of spacetime. The origin of general relativity lies in Einstein’s attempt to apply special relativity in accelerated frames of reference. Remember that the conclusions of relativity were founded for inertial frames, i.e. ones that move only at a uniform velocity. Adding acceleration was a complication that took Einstein 10 years to formulate.

Equivalence Principle:

 equivalence principle equates accelerating and gravity effects The equivalence principle was Einstein’s Newton’s apple’ insight to gravitation. His thought experiment was the following, imagine two elevators, one at rest of the Earth’s surface, one accelerating in space. To an observer inside the elevator (no windows) there is no physical experiment that he/she could perform to differentiate between the two scenarios.The equivalence principle is a fundamental law of physics that states that gravitational and inertial forces are of a similar nature and often indistinguishable. In the Newtonian form it asserts, in effect, that, within a windowless laboratory freely falling in a uniform gravitational field, experimenters would be unaware that the laboratory is in a state of nonuniform motion. All dynamical experiments yield the same results as obtained in an inertial state of uniform motion unaffected by gravity.

 although a simple and common sense assumption, the equivalence principle has strange consequences such as, photons will be effected by gravity, even though they have zero mass An immediate consequence of the equivalence principle is that gravity bends light. To visualize why this is true imagine a photon crossing the elevator accelerating into space. As the photon crosses the elevator, the floor is accelerated upward and the photon appears to fall downward. The same must be true in a gravitational field by the equivalence principle.The principle of equivalence renders the gravitational field fundamentally different from all other force fields encountered in nature. The new theory of gravitation, the general theory of relativity, adopts this characteristic of the gravitational field as its foundation.

 two classical tests of general relativity: the first is the deflection of starlight by the Sun’s gravity as measured by the 1919 solar eclipse experiment There were two classical test of general relativity, the first was that light should bedeflected by passing close to a massive body. The first opportunity occurred during a total eclipse of the Sun in 1919.Measurements of stellar positions near the darkened solar limb proved Einstein was right. Direct confirmation of gravitational lensing was obtained by the Hubble Space Telescope last year.

General Relativity :

 general relativity combines special relativity with the equivalence principle general relativity first resolves the problem of the instantaneous transfer of gravity under Newton’s theory by stating that gravity propagates at the speed of light The second part of relativity is the theory of general relativity and lies on two empirical findings that he elevated to the status of basic postulates. The first postulate is the relativity principle: local physics is governed by the theory of special relativity. The second postulate is the equivalence principle: there is no way for an observer to distinguish locally between gravity and acceleration.The general theory of relativity derives its origin from the need to extend the new space and time concepts of the special theory of relativity from the domain of electric and magnetic phenomena to all of physics and, particularly, to the theory of gravitation. As space and time relations underlie all physical phenomena, it is conceptually intolerable to have to use mutually contradictory notions of space and time in dealing with different kinds of interactions, particularly in view of the fact that the same particles may interact with each other in several different ways–electromagnetically, gravitationally, and by way of so-called nuclear forces. Newton’s explanation of gravitational interactions must be considered one of the most successful physical theories of all time. It accounts for the motions of all the constituents of the solar system with uncanny accuracy, permitting, for instance, the prediction of eclipses hundreds of years ahead. But Newton’s theory visualizes the gravitational pull that the Sun exerts on the planets and the pull that the planets in turn exert on their moons and on each other as taking place instantaneously over the vast distances of interplanetary space, whereas according to relativistic notions of space and time any and all interactions cannot spread faster than the speed of light. The difference may be unimportant, for practical reasons, as all of the members of the solar system move at relative speeds far less than 1/1,000 of the speed of light; nevertheless, relativistic space-time and Newton’s instantaneous action at a distance are fundamentally incompatible. Hence Einstein set out to develop a theory of gravitation that would be consistent with relativity.

 remembering that mass changes with motion, and that mass causes gravity, Einstein links mass, gravity and spacetime with the geometry of spacetime Proceeding on the basis of the experience gained from Maxwell’s theory of the electric field, Einstein postulated the existence of a gravitational field that propagates at the speed of light, c, and that will mediate an attraction as closely as possible equal to the attraction obtained from Newton’s theory. From the outset it was clear that mathematically a field theory of gravitation would be more involved than that of electricity and magnetism. Whereas the sources of the electric field, the electric charges of particles, have values independent of the state of motion of the instruments by which these charges are measured, the source of the gravitational field, the mass of a particle, varies with the speed of the particle relative to the frame of reference in which it is determined and hence will have different values in different frames of reference. This complicating factor introduces into the task of constructing a relativistic theory of the gravitational field a measure of ambiguity, which Einstein resolved eventually by invoking the principle of equivalence.Einstein discovered that there is a relationship between mass, gravity and spacetime. Mass distorts spacetime, causing it to curve. Gravity can be described as motion caused in curved spacetime .

 gravity as geometry of spacetime returns physics to classic levels of the ancient Greeks however, spacetime is not Euclidean matter tells spacetime how to curve, and spacetime tells matter how to move (orbits) Thus, the primary result from general relativity is that gravitation is a purely geometric consequence of the properties of spacetime. Special relativity destroyed classical physics view of absolute space and time, general relativity dismantles the idea that spacetime is described by Euclidean or plane geometry. In this sense, general relativity is a field theory, relating Newton’s law of gravity to the field nature of spacetime, which can be curved.Gravity in general relativity is described in terms of curved spacetime. The idea that spacetime is distorted by motion, as in special relativity, is extended to gravity by the equivalence principle. Gravity comes from matter, so the presence of matter causes distortions or warps in spacetime. Matter tells spacetime how to curve, and spacetime tells matter how to move (orbits).

 the 2nd test was the prediction of time dilation in a gravitational field, first shown by atomic clocks in the mid-70’s (note the need of advanced technology to test general relativity) the effects of general relativity require sensitive instruments under the condition of weak fields, i.e. conditions where the acceleration due to gravity is much, much less than the speed of light strong fields are found in extreme situations such as near neutron stars or black holes The second test is that general relativity predicts a time dilation in a gravitational field, so that, relative to someone outside of the field, clocks (or atomic processes) go slowly. This was confirmed with atomic clocks flying airplanes in the mid-1970’s.The general theory of relativity is constructed so that its results are approximately the same as those of Newton’s theories as long as the velocities of all bodies interacting with each other gravitationally are small compared with the speed of light–i.e., as long as the gravitational fields involved are weak. The latter requirement may be stated roughly in terms of the escape velocity. A gravitational field is considered strong if the escape velocity approaches the speed of light, weak if it is much smaller. All gravitational fields encountered in the solar system are weak in this sense. Notice that at low speeds and weak gravitational fields, general and special relativity reduce to Newtonian physics, i.e. everyday experience.

Black Holes:

 as gravity increases the escape velocity increases when escape velocity exceeds the speed of light a black hole forms The fact that light is bent by a gravitational field brings up the following thought experiment. Imagine adding mass to a body. As the mass increases, so does the gravitational pull and objects require more energy to reach escape velocity. When the mass is sufficiently high enough that the velocity needed to escape is greater than the speed of light we say that a black hole has been created.

 a black hole is still visible by its distortion on local spacetime and the deflection of starlight The Schwarzschild radius marks the point where the event horizon forms, below this radius no light escapes. The visual image of a black hole is one of a dark spot in space with no radiation emitted. Any radiation falling on the black hole is not reflected but rather absorbed, and starlight from behind the black hole is lensed.

 the structure of a black hole contains only an event horizon and a singularity Even though a black hole is invisible, it has properties and structure. The boundary surrounding the black hole at the Schwarzschild radius is called the event horizon, events below this limit are not observed. Since the forces of matter can not overcome the force of gravity, all the mass of a black hole compresses to infinity at the very center, called the singularity.

 the size of a black hole is set by its mass A black hole can come in any size. Stellar mass black holes are thought to form from supernova events, and have radii of 5 km. Galactic black hole in the cores of some galaxies, millions of solar masses and the radius of a solar system, are built up over time by cannibalizing stars. Mini black holes formed in the early Universe (due to tremendous pressures) down to masses of asteroids with radii the size of a grain of sand.

 spacetime is severely distorted near the event horizon, and extreme effects are seen Note that a black hole is the ultimate entropy sink since all information or objects that enter a black hole never return. If an observer entered a black hole to look for the missing information, he/she would be unable to communicate their findings outside the event horizon.

# Quantum Gravity

When it was discovered in the early twentieth century that Newtonian physics, although it had stood unchallenged for hundreds of years, failed to answer basic questions about time and space, such as ‘Is the universe infinite?’ or ‘Is time eternal?’, a new basis for physics was needed.

This lead to the development of Quantum Theory by Bohr, Schrodinger and Heisenberg and Relativity Theory by Einstein. This was the first step in the development of a new basis for physics. Both theories, however are incomplete, and are limited in their abilities to answer many questions. Quantum Physics deals with the behaviour of very small objects, such as atoms, why they do not disintegrate as Newtonian Physics wanted. The theory of Relativity, on the other hand deals with much large scales, celestial bodies and others.

Both theories fail when confronted to the other’s ‘domain’, and are therefore limited in their ability to describe the universe. One must unify these theories, make them compatible with one another. The resulting theory would be able to describe the behavior of the universe, from quarks and atoms to entire galaxies. This is the quantum theory of gravity.

There are two fundamental areas of modern physics, each describes the universe on different scales. First we have quantum mechanics which talks about atoms, molecules and fundamental particles. Then we have general relativity which tells us that gravity is the bending and warping of space-time. There has been much work on finding a theory that combines these two pillars of physics.

There are three main aproches to quantum gravity all have there problems.

1) Loop quantum gravity.
2) String Theory.
3) Others; Penrose spin networks, Connes non-commutative geometry etc.

1) Loop quantum gravity is a way to quantise space time while keeping what General Relativity taught us. It is independent of a background gravitational field or metric. So it should be if we are dealing with gravity. Also, it is formulated in 4 dimensions. The main problem is that the other forces in nature, electromagnetic, strong and weak cannot be included in the formulation. Nor it is clear how loop quantum gravity is related to general relativity.

2) Then we have string theory. String theory is a quantum theory where the fundamental objects are one dimensional strings and not point like particles. String theory is “large enough” to include the standard model and includes gravity as a must. The problems are three fold, first the theory is background dependant. The theory is formulated with a background metric. Secondly no-one knows what the physical vacuum in string theory is, so it has no predictive powers. String theory must be formulated in 11 dimensions, what happened to the other 7 we cannot see? ( Also string theory is supersymmetric and predicts a load of new particles).

3) Then we have other approches, such as non-commutative geometry. This assumes that our space-time coordinates no longer commute. i.e. x y – y x is not zero. This formulation relies heavily on operator algebras.

All the theories have several things in common which are accepted as being part of quantum gravity at about Planck scale.

i)Space-time is discrete and non-commutative ii)Holography and the Bekenstin bound.

i) This is “simply” applying quantum mechanics to space-time. In quantum mechanics all the physical observables are discrete.

ii) The holographic principle was first realised by Hawking. He realised that the entropy of a black hole was proportional to the surface area of the horizon and not the volume. That is all the information about a black hole is on the surface of the horizon. It is like a holograph, you only need to look at the 2-d surface to know everything you can about the black hole.

Bekenstin showed that there is a maximum amount of information that can pass through a surface. It is quantised in Planck units.

# Einstien’s Special Relativity Fully Explained!

Special Theory of Relativity :

 experiments with electromagnetic wave properties of light finds contradictions with Newtonian view of space and time Michelson-Morley experiment shows speed of light is constant regardless of motion of observer (!) By the late 1800’s, it was becoming obvious that there were some serious problems for Newtonian physics concerning the need for absolute space and time when referring to events or interactions (frames of reference). In particular, the newly formulated theory of electromagnetic waves required that light propagation occur in a medium.In a Newtonian Universe, there should be no difference in space or time regardless of where you are or how fast you are moving. In all places, a meter is a meter and a second is a second. And you should be able to travel as fast as you want, with enough acceleration. In the 1890’s, two physicists (Michelson and Morley) were attempting to measure the Earth’s velocity around the Sun with respect to Newtonian Absolute space and time. This would also test how light waves propagated since all waves must move through a medium. For light, this medium was called the aether. The results of the Michelson-Morley experiment was that the velocity of light was constant regardless of how the experiment was tilted with respect to the Earth’s motion. This implied that there was no aether and, thus, no absolute space. Thus, objects, or coordinate systems, moving with constant velocity (called inertial frames) were relative only to themselves. In Newtonian mechanics, quantities such as speed and distance may be transformed from one frame of reference to another, provided that the frames are in uniform motion (i.e. not accelerating).

 Einstein makes constant speed of light key premis to special relativity Considering the results of the Michelson-Morley experiment led Einstein to develop thetheory of special relativity. The key premise to special relativity is that the speed of light (called c = 186,000 miles per sec) is constant in all frames of reference, regardless of their motion. What this means can be best demonstrated by the following scenario:

 special relativity interprets light as a particle called a photon photon moves at speed of light and has zero mass speed of light is an absolute limit, objects with mass must move at less than speed of light This eliminates the paradox with respect to Newtonian physics and electromagnetism of what does a light ray look like’ when the observer is moving at the speed of light. The solution is that only massless photons can move at the speed of light, and that matter must remain below the speed of light regardless of how much acceleration is applied.In special relativity, there is a natural upper limit to velocity, the speed of light. And the speed of light the same in all directions with respect to any frame. A surprising result to the speed of light limit is that clocks can run at different rates, simply when they are traveling a different velocities.

 space and time are variable concepts in relativity time dilation = passage of time slows for objects moving close to the speed of light This means that time (and space) vary for frames of reference moving at different velocities with respect to each other. The change in time is called time dilation, where frames moving near the speed of light have slow clocks.
 Likewise, space is shorten in in high velocity frames, which is called Lorentz contraction

Space-Time Lab

 relativity leads to some strange consequences, such as the twin paradox however, all these predictions have been conferred numerous times by experimentation Time dilation leads to the famous Twins Paradox, which is not a paradox but rather a simple fact of special relativity. Since clocks run slower in frames of reference at high velocity, then one can imagine a scenario were twins age at different rates when separated at birth due to a trip to the stars.It is important to note that all the predictions of special relativity, length contraction, time dilation and the twin paradox, have been confirmed by direct experiments, mostly using sub-atomic particles in high energy accelerators. The effects of relativity are dramatic, but only when speeds approach the speed of light. At normal velocities, the changes to clocks and rulers are too small to be measured.

Spacetime:

 relativity links where and when (space and time) into a 4 dimensional continuum called spacetime position in spacetime are events trajectories through spacetime are called world lines Special relativity demonstrated that there is a relationship between spatial coordinates and temporal coordinates. That we can no longer reference where without some reference to when. Although time remains physically distinct from space, time and the three dimensional space coordinates are so intimately bound together in their properties that it only makes sense to describe them jointly as a four dimensional continuum.Einstein introduced a new concept, that there is an inherent connection between geometry of the Universe and its temporal properties. The result is a four dimensional (three of space, one of time) continuum called spacetime which can best be demonstrated through the use of Minkowski diagrams and world lines.

 determinism is hardened with the concept of spacetime since time now becomes tied to space just as all space is out there’, so is all time Spacetime makes sense from special relativity since it was shown that spatial coordinates (Lorentz contraction) and temporal coordinates (time dilation) vary between frames of reference. Notice that under spacetime, time does not happen’ as perceived by humans, but rather all time exists, stretched out like space in its entirety. Time is simply `there’.

Mass-Energy Equivalence:

 if space and time are variable notions, the momentum must also be relative in order to preserve conservation of energy, mass must be connected to momentum (i.e. energy) Since special relativity demonstrates that space and time are variable concepts, then velocity (which is space divided by time) becomes a variable as well. If velocity changes from reference frame to reference frame, then concepts that involve velocity must also be relative. One such concept is momentum, motion energy.Momentum, as defined by Newtonian, can not be conserved from frame to frame under special relativity. A new parameter had to be defined, called relativistic momentum, which is conserved, but only if the mass of the object is added to the momentum equation. This has a big impact on classical physics because it means there is an equivalence between mass and energy, summarized by the famous Einstein equation:

 mass increases as one nears the speed of light, which explains the limit to the speed of light for material objects, you need infinite acceleration to move an infinitely increasing mass The implications of this was not realized for many years. For example, the production of energy in nuclear reactions (i.e. fission and fusion) was shown to be the conversion of a small amount of atomic mass into energy. This led to the develop of nuclear power and weapons.As an object is accelerated close to the speed of light, relativistic effects begin to dominate. In particular, adding more energy to an object will not make it go faster since the speed of light is the limit. The energy has to go somewhere, so it is added to the mass of the object, as observed from the rest frame. Thus, we say that the observed mass of the object goes up with increased velocity. So a spaceship would appear to gain the mass of a city, then a planet, than a star, as its velocity increased.

 mass-energy equivalence is perhaps the most fundamental discovery of the 20th century photons have momentum, i.e. pressure = solar sails Likewise, the equivalence of mass and energy allowed Einstein to predict that the photon has momentum, even though its mass is zero. This allows the development of light sails and photoelectric detectors.

# SuperSymmetry And Extra Dimensions

Supersymmetry (SUSY) was proposed in the early 1970s as a further symmetry in nature. The Standard Model divides particles into two camps called fermions andbosons. All the usual matter particles we observe – like electrons and quarks – are fermions. Every normal force carrying particle – like a photon or graviton – is a boson.

Roughly speaking, SUSY claims that there’s a way to replace fermions with bosons such that the laws of physics remain the same. Regardless of whether particles are strings or points, SUSY implies a connection between properties of bosonic and fermionic particles.

Supersymmetry tells us that every particle has a partner, which differs in spin by half a unit. All particles have spin. It’s a bit like the rate the Earth rotates on its axis. Spin is an intrinsic quantum mechanical property that does not change. If you change the spin of a photon, it is not a photon any more. Fermions have half-integer spin numbers – ½, 1½, 2½ etc. Bosons have integer spins – 0, 1, 2 etc. The force carriers of strong, weak and electromagnetic forces have spin 1 and the graviton spin 2.

But none of the particles that we ordinarily detect can be partners with each other. Physicists worked out that the new super-partners had to be much heavier than their counterparts, and gave them strange names like squarks, selectrons and photinos. No supersymmetric particles have been discovered so far, but evidence for supersymmetry at particle accelerators like the Large Hadron Collider at CERN would be a landmark for 21st century physics.

Including SUSY makes a big difference to string theory. Supersymmetric string theory (or superstring theory) describes both bosons and fermions, and removes the impossible tachyon (hypothetical particle that always travels faster than the speed of light). Plus it only requires ten dimensions, compared to twenty-six for bosonic string theory. This is a lot closer to the four dimensions we usually experience.All modern work in string theory is based on the superstring. Originally there appeared to be five consistent and distinct superstring theories. It would take a revolution to realise that these were all smoothly connected. They are part of M-theory

Extra dimensions are string theory’s most outlandish prediction. String theory demands that our cosy 4D view of the world is wrong. In fact the universe of strings must have ten dimensions! This is immediately at odds with our perception of reality, but we can resolve the paradox by requiring the six unseen dimensions to be incredibly small.

So what makes a dimension? Intuitively each dimension is an independent direction in which we can move. We live in three dimensions of space, “forward-backward”, “left-right” and “up-down”. There’s also a single time dimension, “past-future”, making 4 dimensions in total. But our perception of dimension is greatly affected by scale.

Imagine watching a faraway ship approaching port. It starts out looking like a zero-dimensional dot on the horizon. Soon you realise it has a mast pointing high into the sky: it now appears to be a one-dimensional line. Next, its sails come into view making it seem two-dimensional. As it nears the dock you finally notice that it has a long deck, the third dimension.

There’s nothing strange here. It’s just that at large distances we can’t resolve dimensions. So perhaps there could be extra dimensions, so small that we don’t perceive them. The process of curling up space to produce these tiny invisible dimensions is known as compactification.

Suppose you’re a squirrel living on an infinitely long tree trunk. The trunk is (more or less) a cylinder. You can move in two independent directions, “along” and “around”.  One day you get bored and move to a thinner tree – the circumference of the trunk is greatly reduced.

Now your “around” dimension is much smaller than it used to be. It only takes a few steps to go all the way round the trunk. Any meaningful movement has to be done in the “along” dimension. You jump to a yet finer trunk. Now a single step takes you round the tree a hundred times! The “around” dimension has become far too small for you to detect.

As the tree trunks get narrower, the dimensions of your world reduce from two to one. In string theory this must happen for all six extra dimensions. We wrap them up so they are inconceivably tiny. Every time you move your hand through space you circle the six hidden dimensions a vast number of times.

The size of these compactified dimensions is similar to the length of a string, the Planck scale. This has two important consequences. Firstly it’s unlikely that we’ll be able to detect them by direct experiment. Nevertheless several possible tests have been suggested, though generally they rely on having a healthy slice of luck. Secondly the extra dimensions form a surface which strings can become caught up in.

The shape and size of strings is vital to modelling their vibrations and interactions. Therefore it’s important to understand how they wrap themselves around the six curled-up dimensions. The precise structure of the surface formed by compactification changes the physics arising from the strings.

It turns out there are many different ways of mushing up the extra dimensions into a tiny space. Which method gives rise to conventional physics? Nobody knows! Current research focuses on Calabi-Yau manifolds, a promising group of compactifications.  But as of yet there is no definitive answer.

# Strings And Particles

We’ve learnt that all strings vibrate as a superposition of modes. Each mode is a particular type of vibration and has an associated energy.

In quantum string theory every mode is identified with a fundamental particle. The equations describing the mode correspond exactly with those defining the particle. For example, the mathematical laws governing photons naturally emerge as the equations for a particular string mode.

This is totally unexpected. String dynamics has nothing to do with electromagnetism, yet Maxwell’s equations appear from nowhere. And that’s not all. The same magic can happens for the other forces of the Standard Model, including the graviton. This is why string theory is a candidate for a fundamental unified theory.

As strings vibrate more and more vigorously, their modes give rise to an infinite number of particles. Einstein’s famous equation E = mc2 tells us that there is an equivalence between energy and mass: the more energetic the string, the heavier the corresponding particle is.

Because fundamental strings are so very small, they form incredibly tight loops and therefore require a colossal amount of energy in order to vibrate. You could pack a million billion billion of these strings into the width of a human hair, but each has the same energy as a train roaring down a track at maximum speed!

So if our strings have such enormous energy, how could they ever correspond to the fundamental particles we observe? Indeed, these have truly tiny masses and thus small energies. Luckily, quantum mechanics comes to the rescue.

Remember the uncertainty principle? It implies that seemingly empty space is filled with energy, called vacuum energy. Physicists worked out that this vacuum energy could cancel with the vibrational energy of the strings, lowering their overall energy and mass.

This cancellation allows vibrating strings to appear as massless, or almost massless, particles. One of string theory’s most celebrated results is that the vibrations of  closed strings automatically give a massless particle with all the properties of the elusive graviton. Gravity emerges naturally from string theory!

Only a few of the string modes would correspond to the particles we see around us. There are infinitely many more particles predicted by string theory, far too heavy to detect in our current laboratories. But there could be indirect ways to look for them. If they were to be discovered it would be a huge leap for string theory.

The first string theories had a few problems. They accounted for bosons (such as force particles like the photon) but did not contain fermions (for example matter particles such as the electron) at all. Secondly the mathematics required twenty-six spacetime dimensions! Lastly the theory predicted an impossible particle with negative mass called the tachyon.

Thankfully supersymmetry solved these difficulties. Superstring theory produces fermions as well as bosons and removes the need for an impossible tachyon. It also brings the required dimensions down to ten – nine space and one time.

# A World Of Strings

Welcome to the theory where everything is made of strings. They’re much as you might imagine from everyday life; strings can wiggle and contort in myriad ways. In doing so they give rise to particles and forces. Remarkably, the motion of a string can encode both particles and forces.

To understand string theory we must study the physics of strings. It’s good to start with a musical example: you’re playing guitar in a rock band. It’s a huge gig and the crowd is expectant. Once the noise has died down you pluck your first string. It begins to move, producing a characteristic musical note.

As you play, your fingers cause other strings to vibrate. Depending on their shape and size they sound at different pitches. The overall effect is a breathtaking display of harmony and rhythm; it’s no wonder you guys are so successful. And all because of some vibrating strings.

The analogy with string theory is immediate. The guitar strings become the fundamental strings of nature. These vibrate in different ways depending on their length and energy. The different musical pitches correspond to individual particles. The harmonies of the band represent interactions between these particles.

It turns out that these fundamental strings are probably incredibly small. The best estimate for a typical string length is 10-33 cm. This length is often called the Planck scale. If you were to lay a thousand billion billion strings end to end you would only just cover the width of a single atom. They are too minuscule to be detected by our current accelerators. This means that we must look for experimental evidence indirectly.

We partition strings into two categories. Open strings have two endpoints. These might be fixed like on a guitar or could be free to move as they please. Fixing the endpoints in particular ways gives rise to distinct vibrational patterns. Closed strings have no endpoints and so form a complete loop.

A rubber band is a good model for a string. Vibrations involve the band stretching and compressing. The tension in the rubber provides the energy to drive the motion. The tiny fundamental strings require huge tension to keep them small. Correspondingly they oscillate with very large energies.

By Einstein’s famous equation E = mc2 we know that energy is equivalent to mass. Hence a high energy string is the same as a very heavy string. The typical mass of a string is astronomical compared to that of a proton. But if strings are so heavy how can they possibly constitute elementary particles? Fortunately quantum corrections sort out the issue.

So how do strings vibrate? Unsurprisingly they undulate like waves. You can easily see this with a skipping rope. Fix one end of the rope and hold the other. By moving your arm quickly you can send a wave along the material. Different movements create complex patterns of several waves added together. Physicists call this phenomenon a superposition of waves.

So a string sways as a superposition of different oscillations. Each constituent vibration is known as a mode. Adding up all the modes gave you the complex dynamics of the skipping rope. The same is true of our microscopic strings. We can now precisely explain how strings produce particles.

# Gravitational Waves,OK. But Where Are Gravitons? (Very Detailed)

## Intro.

Present-day physics cannot describe what happened in the Big Bang. Quantum theory and the theory of relativity fail in this almost infinitely dense and hot primal state of the universe. Only an all-encompassing theory of quantum gravity which unifies these two fundamental pillars of physics could provide an insight into how the universe began. Einstein and his successors, who have been searching for this for almost one hundred years.

Space consists of tiny elementary cells or “atoms of space” in some modern theories of quantum gravity trying to unify General Relativity and Quantum Mechanics. Quantum gravity should make it possible to describe the evolution of the universe from the Big Bang to today within one single theory.Our world is ruled by four fundamental forces: the gravitational pull of massive objects, the electromagnetic interaction between electric charges, the strong nuclear interaction holding atomic nuclei together and the weak nuclear force causing unstable ones to fall apart. Physicists have quantum theories for the last three of them that allow very precise calculations of phenomena on the smallest, subatomic scales. However, gravity does not fit into this scheme. Despite decades of research, there is no generally accepted quantum theory of gravity, which is needed to better understand fundamental aspects of our universe.

## Gravitons.

Gravitons are tiny particles that carry the “force” of gravity. They are what brings you back down to Earth when you jump. So why have we never seen them, and why are they so impossibly complicated we need string theory to figure them out?

Even without observing gravitons, scientists know a few things about them. They know, because gravity is a force with an infinite reach, that gravitons would have to be massless. This technically makes them “gauge bosons,” and puts them in the company of photons and gluons. Scientists also know that gravitons have a spin of two, which makes them unique among particles. The combined properties mean that, if scientists were able to pin down an event involving a mysterious particle with no mass and a spin of two, they would know they were looking at a graviton.

## Where String Theory Jumps In.

In our everyday lives, we experience three spatial dimensions, and a fourth dimension of time. How could there be more? Einstein’s general theory of relativity tells us that space can expand, contract, and bend. Now if one dimension were to contract to a size smaller than an atom, it would be hidden from our view. But if we could look on a small enough scale, that hidden dimension might become visible again. Imagine a person walking on a tightrope. She can only move backward and forward; but not left and right, nor up and down, so she only sees one dimension. Ants living on a much smaller scale could move around the cable, in what would appear like an extra dimension to the tightrope-walker.

How could we test for extra dimensions? One option would be to find evidence of particles that can exist only if extra dimensions are real. Theories that suggest extra dimensions predict that, in the same way as atoms have a low-energy ground state and excited high-energy states, there would be heavier versions of standard particles in other dimensions. These heavier versions of particles – called Kaluza-Klein states – would have exactly the same properties as standard particles (and so be visible to our detectors) but with a greater mass. If CMS or ATLAS were to find a Z- or W-like particle (the Z and W bosons being carriers of the electroweak force) with a mass 100 times larger for instance, this might suggest the presence of extra dimensions. Such heavy particles can only be revealed at the high energies reached by the Large Hadron Collider (LHC).

Why is gravity so much weaker than the other fundamental forces? A small fridge magnet is enough to create an electromagnetic force greater than the gravitational pull exerted by planet Earth. One possibility is that we don’t feel the full effect of gravity  because part of it spreads to extra dimensions. Though it may sound like science fiction, if extra dimensions exist, they could explain why the universe is expanding faster than expected, and why gravity is weaker than the other forces of nature.

There is, however, a major problem. To understand it, let’s go back to photons and electrons. When an electron falls from one level to another, out pops a photon. When that photons falls, or otherwise moves, it produces no second photon. Electron movement produces photons. Photon movement does not produce more photons. There are occasional times when photons can do odd things. They can split into electron and positron pairs, which can produce more photons, and which then recombine into a photon again. Although this burst of particles may get hectic, it doesn’t produce an endless branching chain of photons. Because of this, photons and electron interactions are said to be renormalizable. They can get weird, but they can’t become endless.

Gravitons are not so tame. While photons are spawned by movement in electrons, gravitons are whelped by energy and mass. Gravitons are massless, but they do carry energy. This means a graviton can create more gravitons.

Like other quantum particles, gravitons can carry a lot of energy, or momentum, when confined to a small space. A graviton is confined to a small space when one graviton is popping out another graviton. At that moment, two gravitons are in a tiny space, one right next to the other. That huge amount of energy causes the newly-created graviton to create yet another graviton. This endless cycle of graviton production makes gravitons nonrenormalizable.

String theory is invoked in these situations part because nonrenormalizable gravitons are points. Strings are longer than points, and so the creation of the stringy graviton isn’t so confined in time and space. That bit of wiggle room keeps the creation of a graviton from being so energetic that it necessitates the creation of yet another graviton, and makes the theory renormalizable.

A graviton is really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really really hard to detect.

## The LIGO detector and team??

Quantum gravity is a theory that has been the target of decades of study by physicists worldwide. If this idea is proven, it would tie together the General Theory of Relativity (which governs gravitational fields) with quantum mechanics, and the bizarro-world of subatomic particles.

Gravitational waves, produced by accelerating objects, ripple through space-time, according to most interpretations of the General Theory of Relativity penned by famed physicist Albert Einstein. Researchers at the Laser Interferometer Gravitational-Wave Observatory (LIGO) have announced they detected these disturbances in the fabric of time and space for the first time.

Certain aspects of subatomic behavior is quantized – they can only move or exist in particular whole number states. This characteristic may be thought of like steps to walk up into an apartment. Many physicists believe gravitational waves are similarly quantized, made up of individual quantum particles of gravity – gravitons.

Although it is not certain, many physicists believe that these particles join together, forming the gravitational waves that travel through space. Like photons of light, these gravitons, would have no rest mass, and move at the speed of light.

The effects of quantum gravity are predicted to be quite pronounced in the region immediately surrounding the center of black holes. However, it is impossible to collect data from events near a singularity. The events witnessed by astronomers are LIGO-recorded activity from just outside the event horizons of a pair of black holes as they collided.

The LIGO detector cannot detect single gravitons, and cannot, by itself, test the theory of quantum gravity. However, there is reason to believe that either LIGO, or a future gravitational wave detector, could be used to find evidence of quantum gravity by examining the emission spectrum of energy seen surrounding the event horizons of black holes.

According to some theories, even at the event horizon, the effects of gravitons could cause gravitational waves to be more powerful, and less regular, than they would be without their influence.

“Certain scenarios with strong quantum modifications in a region extending well outside the horizon are expected to modify classical evolution, and distort the near-peak gravitational wave signal, suggesting a search for anomalies such as decreased regularity of the signal and increased power,” Steven Giddings of the University of California, Santa Barbara, said.

Any variation seen between observations and graviton-free theories of gravitational waves, such as Einstein’s, could assist physicists seeking to understand the ultimate units of gravity.

As astronomers use LIGO and other detectors to search for elusive ripples in space-time, they may also come across evidence of other strange features of space, including cosmic strings, theoretical one-dimensional strings of energy, which may have been created long ago, when the universe was young.

Moreover, as additional findings of gravitational waves are recorded, physicists will search the data for behavior of the ripples which might suggest the presence of gravitons. If they are found, the discovery could herald a new age of understanding how gravity works. Such a finding could suggest that other notions of gravity, such as string theory, could prove to be the basis of future work on the nature of gravity.

But until such variations are seen, the existence of gravitons remains strictly theoretical.

## A Small Packet Of Gravity?

Some theorists suggest that a particle called the “graviton” is associated with gravity in the same way as the photon is associated with the electromagnetic force. If gravitons exist, it should be possible to create them at the LHC, but they would rapidly disappear into extra dimensions. Collisions in particle accelerators always create balanced events – just like fireworks – with particles flying out in all directions. A graviton might escape our detectors, leaving an empty zone that we notice as an imbalance in momentum and energy in the event. We would need to carefully study the properties of the missing object to work out whether it is a graviton escaping to another dimension or something else. This method of searching for missing energy in events is also used to look for dark matter or supersymmetric particles.

## Quantum(Microscopic) Black Holes

Another way of revealing extra dimensions would be through the production of “microscopic black holes”. What exactly we would detect would depend on the number of extra dimensions, the mass of the black hole, the size of the dimensions and the energy at which the black hole occurs. If micro black holes do appear in the collisions created by the LHC, they would disintegrate rapidly, in around 10-27 seconds. They would decay into Standard Model or supersymmetric particles, creating events containing an exceptional number of tracks in our detectors, which we would easily spot. Finding more on any of these subjects would open the door to yet unknown possibilities.

## Photons easy to detect…So why not gravitons?

You have to appreciate just how weak gravity really is. Gravity is 10^-36 times weaker than the electromagnetic force. Trying to detect individual gravitons will be impossible with the tools we have at hand. That is why scientists are trying to detect gravity waves generated during the early stages of the universe. The effect of gravity on mass is independent of the size of the mass unless the masses involved are of roughly equivalent size. If one mass is very much smaller than the other then its influence is negligible. The tides, whilst involving a substantial amount of mass are a very bad choice. Firstly, they are composed of liquids which are not only reacting to the gravitational force, but to the kinetic energy of the molecules that constitute the oceans. Secondly, overall they compose only a fraction of the mass of the whole earth.

Detecting a photon, for example, is extremely easy. There many types of devices that are able to detect single photons, such as photomultipliers, used in labs around the world. In fact, you don’t even need any fancy technology; the human eye can, in principle, detect a single photon. (that isnt quite the topic so i wouldnt go into depths)

## Possible Way To Detect A Graviton

However, detecting gravitons is much (much much etc…) harder. A famous example  considers an ideal detector with the mass of the planet Jupiter, around 10271027 kilograms, placed in close orbit around a neutron star, which is a very strong source of gravitons. A back-of-the-envelope calculation reveals that even in this extremely unrealistic scenario, it would take 100 years to detect a single graviton!

Okay, you say, so let’s just make that detector (sometime in the far future when we have the technology to do so) and wait for 100 years. There’s a crucial detail that I forgot to mention, however. The star also emits neutrinos in addition to gravitons; in fact, many more neutrinos than gravitons. And neutrinos are much easier to detect than gravitons. In fact, we can calculate that for every graviton that is detected in this scenario, around 10331033neutrinos will be detected. So we will never be able to find the one graviton among the 10331033 neutrinos.

Ah, you say, but we can build a neutrino shield and block the neutrinos! But such a shield would need to have a thickness of several light years, and if you try to make it more dense in order to fit between the star and the detector, it would collapse into a black hole…

In conclusion, even with insanely advanced futuristic technology, it would simply be impossible to detect a graviton.

What we have been able to detect, though, are gravitational waves. This amazing discovery by the LIGO experiment was announced on February 11 2016. Gravitational waves are made of lots and lots of gravitons, just like electromagnetic waves are made of lots and lots of photons. A typical gravitational wave is composed of roughly 1,000,000,000,000,000 gravitons per cubic centimeter, therefore it is obviously much easier to detect than a single graviton.

On the other hand, we definitely do not have the technology to detect individual gravitons, and unless some new ingenious way to detect them is found, we will never be able to do so even with much more advanced technology.

What are the consequences of this technological impossibility to detect gravitons? As it turns out, it doesn’t really matter! Let me explain.

First, where exactly do gravitons appear in physics? Theoretical physicists are trying to combine general relativity and quantum mechanics into a single theory, called quantum gravity. We do not have a final theory of quantum gravity yet, but we are working very hard on it, and we already understand many aspects of what such a theory should be.

In a theory of quantum gravity, gravitons are the quanta of the gravitational field. Therefore, quantum gravity will use gravitons as part of its formulation, just like the theory of quantum electrodynamics uses photons, which are the quanta of the electromagnetic field.

However, we did not confirm quantum electrodynamics experimentally by detecting photons. Quantum electrodynamics produces predictions that are different from those of classical electrodynamics, and by experimentally testing these predictions we have been able to confirm that the electromagnetic field is indeed quantized.

In a similar way, when we finally have a good candidate for a theory of quantum gravity, it will produce predictions that are different from those of classical gravity. By experimentally testing these predictions, we will be able to confirm that the gravitational field is quantized.

In other words, what we need to do is not detect gravitons; we need to test the predictions of a theory of quantum gravity, as soon as we have such a theory. This will indirectly confirm the existence of gravitons.

## Credits to:

Me (Mainly)

LHC website

Gizmodo website for some part of photons info.

🙂 Hope u enjoyed