Special Theory Of Relativity

Special relativity (SR) or the '''special theory of relativity''' is a Physical theory originally formulated in a series of papers published between 1895 and 1905 by Hendrik Lorentz and Henri Poincaré and again, most popularly, in 1905 by Albert Einstein in " On The Electrodynamics Of Moving Bodies ". The Special Theory Of Relativity replaced Newtonian Notions Of Space And Time and incorporated Electromagnetism as represented by Maxwell's Equations . The theory is called "special" because it applies the Principle Of Relativity only to the "restricted" or "special" case of inertial reference frames in Flat Spacetime and accelerated frames, where the effects of Gravity can be ignored. Special relativity inter-relates space and time in such a way that the speed of light (as well as other Fundamental Constants ) is constant, and the theory leads to situations where two observers can disagree over Time Interval s and Distance s between events, but without ever disagreeing about what events actually happened. It shows that time can pass more slowly if an observer is moving, depending on their relative speed. The theory also predicts the famous equation E=mc&² . ''For history and motivation, see the article:'' History Of Special Relativity POSTULATES ''Main article: Postulates Of Special Relativity '' # First Postulate - Principle Of Relativity - The laws of physics (including electrodynamics, like propagation of Light ) are the same in all Inertial Frames Of Reference . # Second postulate - Invariance of ''c'' - In empty space, Light always propagates with a constant velocity c, independent of the state of motion of the emitting body. Most current textbooks mistakenly include a major ''derived'' result, that the speed of light is independent of the state of motion of the observer measuring it, as part of the second postulate. A careful reading of Einstein's 1905 paper on this subject shows that, in fact, he made no such assumption. The power of Einstein's argument stems from the manner in which he derived startling and seemingly implausible results from two simple assumptions. One of the most highly counterintuitive of these results (and, as stated above, commonly included in statements of the second postulate), is that the Speed Of Light in Vacuum , commonly denoted ''c'', is the same to all inertial observers. An observer attempting to measure the speed of light's propagation will get the same answer no matter how the observer or the system's components are moving. LACK OF AN ABSOLUTE REFERENCE FRAME The , indicated that the Earth was always 'stationary' relative to the Aether — something that was difficult to explain, since the Earth is in orbit around the Sun. For many, the most elegant solution was to discard the notion of Aether and an absolute frame, and to adopt Einstein's postulates. Special Relativity is formulated so as to not assume that any particular frame of reference is special; rather, in relativity, a system appears to observe the same laws of physics independent of an observer's velocity with respect to it. In particular, the speed of light is always measured to be c, even when measured by multiple observers that are moving at different (but constant) velocities. No matter how fast the object travels, the relative speed of light for the object is always c. CONSEQUENCES ''Main article: Consequences Of Special Relativity '' Einstein has said that all of the consequences of special relativity can be found from examination of the Lorentz Transformations . These transformations, and hence Special Relativity, lead to different physical predictions than Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counter-intuitive: Time dilation - the time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames. (e.g., the Twin Paradox which concerns a twin who flies off in a spaceship travelling near the speed of light and returns to discover that his twin has aged much more rapidly.) Lack of simultaneity - two events that occur simultaneously to one observer may occur at different times to another observer (lack of Absolute Simultaneity ). Lorentz contraction - the dimensions (e.g., length) of an object as measured by one observer may be smaller from the results of measurements of the same object made by another observer. (e.g., the Ladder Paradox involves a long ladder travelling near the speed of light and being contained within a smaller garage.) Addition of velocities - speeds do not simply add, for example if a rocket is moving at 2/3 the speed of light relative to an observer, and the rocket fires a missile at 2/3 of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. Mass and momentum - when gaining momentum the apparent mass of an object increases as well as the energy (giving the famous E=mc² equation.) THE LORENTZ TRANSFORMATIONS OF SPACE AND TIME : ''Full article: Lorentz Transformation s of a rapidly accelerating observer. In this animation, the vertical direction indicates time and the horizontal direction indicates distance, the dashed line is the spacetime trajectory (" World Line ") of the observer. The lower quarter of the diagram shows the events that are visible to the user, and the upper quarter shows the Light Cone - those that will be able to see the observer. The small dots are arbitrary events in spacetime. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates.]] Relativity theory depends on " Reference Frame s". A reference frame is a point in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes. In addition, a reference frame has a clock, moving with the reference frame allowing the measurement of the time of events. An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: It is a "point" in Space-time . Since the speed of light is constant in Relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired. For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four space-time coordinates: The time of occurrence and its 3-dimensional spatial location from a reference point. Let's call this reference frame $S$ . In relativity theory we often want to calculate the position of a point from a different reference point. Suppose we have a second reference frame $S'$ , whose spatial axes and clock exactly coincide with that of $S$ at time zero, but it is moving at a constant velocity $v$ with respect to $S$ along the $x$ axis. Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be ''comoving''. Therefore $S$ and $S'$ are not ''comoving''. Let's define the Event to have space-time coordinates $(t, x, y, z)$ in system $S$ and $(t', x', y', z')$ in $S'$ . Then the Lorentz transformation specifies that these coordinates are related in the following way (which is the result of Rotation of space-time): : $t' = \gamma \left(t - rac{v x}{c^{2}} ight)$ : $x' = \gamma (x - v t)\,$ : $y' = y\,$ : $z' = z\,$ where $\gamma \equiv rac{1}{\sqrt{1 - v^2/c^2}}$ is called the Lorentz Factor and $c$ is the Speed Of Light in a vacuum. The $y$ and $z$ coordinates are unaffected, but the $x$ and $t$ axes are mixed up by the transformation. In fact, they are a form of rotation. A quantity invariant under Lorentz transformations is known as a Lorentz Scalar . LORENTZ CONTRACTION AND TIME DILATION Multiplying through we get: : $t' = \gamma t - \gamma rac{v x}{c^{2}}$ : $x' = \gamma x - \gamma v t\,$ Examination of the first term of the equation for $x'$ in the Lorentz transformation shows that all positions $x$ in one frame are multiplied by gamma, a number greater than one, to calculate the spatial interval in the second non comoving frame. This may be correctly interpreted as a physical contraction of any object from full sized, and at rest in one frame, to the second frame in which it is moving. This is termed ''Lorentz Contraction''. Similarly, in the equation for time $t'$ , $t$ is multiplied by gamma in the second frame. This may be interpreted as time physically proceeding more slowly when an object is moving than in the rest frame of the object. This is termed ''Time Dilation''. It might be expected that since one frame seems contracted, from the contracted frame of reference, the other would seem expanded; and similar effects with time. However since the Lorentz equations are symmetrical with respect to opposite relative speed, each frame ''seemingly'' paradoxically sees the other as equally contracted and equally time dilated. These effects are not merely appearances; the time in the different frame of references essentially do travel at different rates to each other and the lengths of objects really are physically changed whilst in relative motion. See also the Twin Paradox . SIMULTANEITY Special relativity holds that events that are Simultaneous in one frame of reference need not be simultaneous in another frame of reference. Simultaneity can be seen by considering the second term of the expanded Lorentz equation for $t'$ . Here as the velocity $v$ varies two events move forwards or backwards in time relative to each other if they are physically separated in space. This can be observed in Diagram 1; some events may be observed moving from the past to the future and back again as acceleration between reference frames occurs and time passes. Lack of simultaneity implies that, for example, the two ends of a moving rod actually are not equally old — so for example, a cast radioactive rod would be older and have lower activity at the trailing edge than the leading edge. Indeed, lack of simultaneity explains why Lorentz contraction occurs — the rod is partially tilted along the time axis as it accelerates, giving a foreshortening in the spatial dimension. According to a paper by Los Alamos scientist James Terrell, the observability of the Lorentz contraction from the single point in space is impossible by optical instruments. For example a distant spherical galaxy moving across the Milky Way with velocity 0.99c would appear as spherical object in our optical instruments. Lorentz contraction of distant galaxies cannot be observed from Earth. see "Invisibility of the Lorentz Contraction" in Physical Review 116 (1959) 1041 . Lorentz contraction is nevertheless a real physical effect. Its measurement by widely spaced instruments used simultaneously in the given frame is possible. CAUSALITY In diagram 2 the interval AB is 'time-like'; ''i.e.'', there is a frame of reference in which event A and event B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect). The interval AC in the diagram is 'space-like'; ''i.e.'', there is a frame of reference in which event A and event C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. Barring some way of traveling Faster Than Light , it is not possible for any matter (or information) to travel from A to C or from C to A. Thus there is no direct causal connection between A and C. However, many points in spacetime would be in the light cone of both C and A and can be causally related to either or both of these events, and similarly both C and A could have been caused by an earlier event. Since the set of points of spacetime that is in any events light cone is completely independent of reference frame, then causality is absolutely assured. ADDITION OF VELOCITIES If the observer in $S$ sees an object moving along the $x$ axis at velocity $w$ , then the observer in the $S'$ system will see the object moving with velocity $w'$ where : $w'=rac{w-v}{1-wv/c^2}$ . This equation can be derived from the space and time transformations above. Notice that if the object is moving at the speed of light in the $S$ system (i.e. $w=c$ ), then it will also be moving at the speed of light in the $S'$ system. Also, if both $w$ and $v$ are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities: $w'=w-v$ . MASS, MOMENTUM, AND ENERGY In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of Mass , Momentum , and Energy , all of which are important constructs in Newtonian Mechanics . Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated. There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses Conservation Law s. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple Thought Experiment s using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR. Given an object of Invariant Mass ''m''₀ traveling at velocity ''v'' the energy and momentum are given by : $E = \gamma m_0 c^2 \,\!$ : $ec p = \gamma m_0 ec v \,\!$ where ''γ'' (the Lorentz Factor ) is given by : $\gamma = rac{1}{\sqrt{1 - v^2/c^2}} \,\!$ and ''c'' is the speed of light. The term γ occurs frequently in relativity, and comes from the Lorentz Transformation Equations . Relativistic energy and momentum can be related through the formula : $E^2 - (p c)^2 = (m_0 c^2)^2 \,\!$ which is referred to as the ''relativistic energy-momentum equation''. For velocities much smaller than those of light, γ can be approximated using a Taylor Series Expansion and one finds that : $E \approx m_0 c^2 + \begin{matrix} rac{1}{2} \end{matrix} m_0 v^2 \,\!$ : $ec p \approx m_0 ec v \,\!$ Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian Kinetic Energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities. Looking at the above formulas for energy, one sees that when an object is at rest (''v'' = 0 and γ = 1) there is a non-zero energy remaining: : $E = m_0 c^2 \,\!$ This energy is referred to as ''rest energy''. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which are meaningful. Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity: ''Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy.'' {Link without Title} This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by Nuclear Reaction s, providing important information which was useful in the development of the Nuclear Bomb . The implications of this formula on 20th Century life have made it one of the most famous equations in all of science. RELATIVISTIC MASS Introductory physics courses and some older textbooks on special relativity sometimes define a '' Relativistic Mass '' which increases as the velocity of a body increases. According to the geometric interpretation of special relativity, this is often deprecated and the term 'mass' is reserved to mean 'rest mass' and is thus independent of the inertial frame, i.e., ''invariant''. Using the relativistic mass definition, the mass of an object may vary depending on the observer's inertial frame in the same way that other properties such as its length may do so. Defining such a quantity may sometimes be ''useful'' in that doing so simplifies a calculation by restricting it to a specific frame. For example, consider a body with an invariant mass m₀ moving at some velocity relative to an observer's reference system. That observer defines the ''relativistic mass'' of that body as: : $m = \gamma m_0\!$ "Relativistic mass" should not be confused with the "longitudinal" and "transverse mass" definitions that were used around 1900 and that were based on an inconsistent application of the laws of Newton: those used ''F=ma'' for a variable mass, while relativistic mass corresponds to Newton's dynamic mass in which ''p=mv'' and ''F=dp/dt''. Note also that the body does ''not'' actually become more massive in its ''proper'' frame, since the relativistic mass is only different for an observer in a different frame. The ''only'' mass that is frame independent is the invariant mass. When using the relativistic mass, the used reference frame should be specified if it isn't already obvious or implied. It also goes almost without saying that the increase in relativistic mass does not come from an increased number of atoms in the object. Instead, each atom, indeed each subatomic particle increases its relativistic mass as the object accelerates. Physics textbooks sometimes use the relativistic mass as it allows the students to utilize their knowledge of Newtonian physics to gain some intuitive grasp of relativity in their frame of choice (usually their own!). "Relativistic mass" is also consistent with the concepts "time dilation" and "length contraction". FORCE The classical definition of force F, : $ec F = dec p/dt$ is valid in relativity. Force is the time-derivative of momentum, so the classical formula for second Newton law has to be replaced by : $ec F = \gamma m_0 ec a + \gamma^3 m_0 rac{ec v \cdot ec a}{c^2} ec v$ As seen from the equation, force and acceleration vectors are not necessarily parallel in relativity. THE GEOMETRY OF SPACE-TIME SR uses a 'flat' 4-dimensional Minkowski Space , which is an example of a Space-time . This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with. The Differential of distance(''ds'') in cartesian 3D space is defined as: : $ds^2 = dx_1^2 + dx_2^2 + dx_3^2$ where $(dx_1,dx_2,dx_3)$ are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, is added, with units of C , so that the equation for the differential of distance becomes: : $ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2$ To make time coordinate symmetric to space coordinates, we must therefore treat time as Imaginary : dx₄ = ict . In this case the above equation becomes symmetric: : $ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$ This is, however, not just a simplifying mathematical technique, but has profound theoretical significance as it shows that special relativity is simply a Rotational Symmetry of our Space-time , very similar to rotational symmetry of Euclidean Space . The deeper development of the theory relies on the concept of the Minkowski metric as described below. If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space : $ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2$ We see that the Null Geodesic s lie along a dual-cone: defined by the equation : $ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2$ or : $dx_1^2 + dx_2^2 = c^2 dt^2$ dt''. If we extend this to three spatial dimensions, the null geodesics is 4-dimensional cone: : $ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2$ : $dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2$ This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old.", we are looking down this line of sight: a null geodesic. We are looking at an event $d = \sqrt{x_1^2+x_2^2+x_3^2}$ meters away and ''d/c'' seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".) The cone in the ''-t'' region is the information that the point is 'receiving', while the cone in the ''+t'' section is the information that the point is 'sending'. The geometry of Minkowski space can be depicted using Minkowski Diagram s, which are also useful in understanding many of the thought-experiments in special relativity. PHYSICS IN SPACETIME Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any reference frame) as : $\eta^{\alpha\beta} =\eta_{\alpha\beta} = \begin{pmatrix} -1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 \end{pmatrix}.$ Then we recognise that co-ordinate transformations between inertial reference frames are given by the Lorentz Transformation tensor Λ. For the special case of motion along the x-axis, we have: : $\Lambda^\mu{}_ u = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0\ -\beta\gamma & \gamma & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$ where β and γ are defined as : $\beta = rac{v}{c}, \gamma = rac{1}{\sqrt{1-\beta^2}}.$ This simplifies almost every formula ever encountered in special relativity. We understand that all proper physical quantities are given by tensors. So to transform from one frame to another, we use the well known Tensor Transformation Law : $T^{\left[i_1',i_2',...i_p' ight]}_{\left[j_1',j_2',...j_q' ight]} = \Lambda^{i_1'}{}_{i_1}\Lambda^{i_2'}{}_{i_2}...\Lambda^{i_p'}{}_{i_p} \Lambda_{j_1'}{}^{j_1}\Lambda_{j_2'}{}^{j_2}...\Lambda_{j_q'}{}^{j_q} T^{\left[i_1,i_2,...i_p ight]}_{\left[j_1,j_2,...j_q ight]}.$ To see how this is useful, we first recognise that position is a four vector, since in component form : $x_ u=\left(-ct, x, y, z ight).$ So to transform it from an unprimed co-ordinate system ''S'' to a primed system ''S''', we calculate : $\begin{pmatrix} ct'\ x'\ y'\ z' \end{pmatrix} = x'^{\mu}=\Lambda^\mu{}_ u x^ u= \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0\ -\beta\gamma & \gamma & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct\ x\ y\ z \end{pmatrix} = \begin{pmatrix} \gamma ct- \gamma\beta x\ \gamma x - \beta \gamma ct \ y\ z \end{pmatrix}.$ which is a nice way of arriving at the cumbersome looking definition of the Lorentz transformation given above. But the true power becomes evident when you recognise that all tensors transform by the same rule. Firstly, note that the scalar (in fact the length of the position four-vector) constructed as follows:
	Is Invariant - Ie It Takes The Same Value In All Inertial Frames, Simply Because It Is A 0 Rank Tensor, And So No Copies Of The Lorentz Appears In Its Transformation: x'	x
	:<math>p	p^\mu p_\mu = -E^2 + (pc)^2\,</math>
	:<math>p	p^\mu p_\mu = -Erest^2\,</math>
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Special Theory Of Relativity