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Message: <blockquote data-quote="mutantlast" data-source="post: 2941421" data-attributes="member: 115180">[edit] Force The classical definition of ordinary force f is given by Newton's Second Law in its original form: <img src="http://upload.wikimedia.org/math/a/0/e/a0e352ec4c36fe245d8e8f12aefae9b1.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> and this is valid in relativity. Many modern textbooks rewrite Newton's Second Law as <img src="http://upload.wikimedia.org/math/b/a/d/bad7a3dfa148b8d91460887479cdc192.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> This form is not valid in relativity or in other situations where the relativistic mass M is varying. This formula can be replaced in the relativistic case by <img src="http://upload.wikimedia.org/math/6/c/3/6c3adfb901ca0e8b78ad13a65731c57c.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> As seen from the equation, ordinary force and acceleration vectors are not necessarily parallel in relativity. However the four-vector expression relating four-force <img src="http://upload.wikimedia.org/math/6/b/5/6b5c9f34a867390e015a73de6b4c7768.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> to invariant mass m and four-acceleration <img src="http://upload.wikimedia.org/math/8/5/d/85dad09955d0591fe6b69159d8855327.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> restores the same equation form <img src="http://upload.wikimedia.org/math/c/c/2/cc2b126adb7a9ccdc2800d87e218bcac.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />  [edit] The geometry of space-time  Main article: Minkowski space 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: <img src="http://upload.wikimedia.org/math/d/5/3/d531b51aecf3401925695a3a31fffdb9.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> where (dx1,dx2,dx3) are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes: <img src="http://upload.wikimedia.org/math/2/f/1/2f1acb8b2fdd9eea4b76ad0323912c40.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary: x4 = ict . In this case the above equation becomes symmetric: <img src="http://upload.wikimedia.org/math/8/8/d/88dcdf91e920635fcfc4b47aca5625e5.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space. Just as Euclidean space uses a Euclidean metric, so space-time uses a Minkowski metric. Basically, SR can be stated in terms of the invariance of space-time interval (between any two events) as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski space-time. According to Misner (1971 §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) rather than a "disguised" Euclidean metric using ict as the time coordinate. If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space <img src="http://upload.wikimedia.org/math/d/a/e/dae91985413e63aefc6e7aa08b7e16b1.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> We see that the null geodesics lie along a dual-cone: <img src="http://upload.wikimedia.org/wikipedia/en/thumb/4/4f/Sr1.svg/134px-Sr1.svg.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> defined by the equation <img src="http://upload.wikimedia.org/math/a/9/1/a9114f064423d7f47a2fc95adba3b8d0.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> or <img src="http://upload.wikimedia.org/math/7/9/f/79f7cdedabf679384a8d9bb166eec55b.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Which is the equation of a circle with r=c×dt. If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone: <img src="http://upload.wikimedia.org/wikipedia/commons/c/cd/Sr3.jpg" alt="" class="fr-fic fr-dii fr-draggable " style="" /> <img src="http://upload.wikimedia.org/math/c/7/2/c72e1099f9fcceac1978f1cd3b3d1862.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> <img src="http://upload.wikimedia.org/math/9/8/4/98496bb56c45c8fb5213166671632065.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> 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 <img src="http://upload.wikimedia.org/math/6/0/f/60f9a792ce08526a5d12f5df47eebf81.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> 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".) <img src="http://upload.wikimedia.org/wikipedia/en/thumb/4/4f/Sr1.svg/134px-Sr1.svg.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> 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 diagrams, which are also useful in understanding many of the thought-experiments in special relativity.  [edit] Physics in spacetime Here, we see how to write the equations of special relativity in a manifestly Lorentz covariant form. The position of an event in spacetime is given by a contravariant four vector whose components are: <img src="http://upload.wikimedia.org/math/2/0/b/20bf8ba56d3be4008a6a2952d9a437a2.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> That is, x0 = t and x1 = x and x2 = y and x3 = z. Superscripts are contravariant indices in this section rather than exponents except when they indicate a square. Subscripts are covariant indices which also range from zero to three as with the spacetime gradient of a field φ: <img src="http://upload.wikimedia.org/math/0/3/5/035de9ff524fa8351c1b67a736b5ccc8.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />  [edit] Metric and transformations of coordinates Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any inertial reference frame) as: <img src="http://upload.wikimedia.org/math/a/5/6/a56bbc8865c96d757222af4927f7c91a.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Its reciprocal is: <img src="http://upload.wikimedia.org/math/a/e/e/aee78a5d8bc51896cb05ba464cfce283.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Then we recognize that coordinate 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: <img src="http://upload.wikimedia.org/math/2/e/b/2eb20e96de68a1847b4c14c7f8308596.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> which is simply the matrix of a boost (like a rotation) between the x and t coordinates. Where μ' indicates the row and ν indicates the column. Also, β and γ are defined as: <img src="http://upload.wikimedia.org/math/0/5/d/05dbd54b210913ced0694b00b6c68e6f.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy: <img src="http://upload.wikimedia.org/math/9/4/e/94eac545386aa4b3fffe2079fd843d5f.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> where there is an implied summation of <img src="http://upload.wikimedia.org/math/f/d/f/fdfbfd67ccba1c81ca34f983664a067b.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> and <img src="http://upload.wikimedia.org/math/b/b/c/bbc801460b176072342e9bed0ab494bd.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> from 0 to 3 on the right-hand side in accordance with the Einstein summation convention. The Poincaré group is the most general group of transformations which preserves the Minkowski metric and this is the physical symmetry underlying special relativity. All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law <img src="http://upload.wikimedia.org/math/5/f/1/5f10063892eef437b2a03d2a0fa9d05d.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Where <img src="http://upload.wikimedia.org/math/1/2/1/121820c3f09f47133fa2f786215f8ee7.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is the reciprocal matrix of <img src="http://upload.wikimedia.org/math/3/b/9/3b9e81ec56f5ecc1fd839b9272259fa2.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />. To see how this is useful, we transform the position of an event from an unprimed coordinate system S to a primed system S', we calculate <img src="http://upload.wikimedia.org/math/c/8/0/c8016c400d6810228d972b8fd919d77b.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> which is the Lorentz transformation given above. All tensors transform by the same rule. The squared length of the differential of the position four-vector <img src="http://upload.wikimedia.org/math/f/d/7/fd7c72e78cf2a0fbe27e9ef243517c17.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> constructed using <img src="http://upload.wikimedia.org/math/2/f/9/2f961a078b51ab468af91c43ad3ebec6.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is an invariant. Being invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. Notice that when the line element <img src="http://upload.wikimedia.org/math/5/4/e/54e29c41cc5f8a4b4df7ee5736ff144e.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is negative that <img src="http://upload.wikimedia.org/math/0/1/2/0122e9b396f8d82c11a9488a242dde91.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is the differential of proper time, while when <img src="http://upload.wikimedia.org/math/5/4/e/54e29c41cc5f8a4b4df7ee5736ff144e.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is positive, <img src="http://upload.wikimedia.org/math/5/e/e/5ee4ea10ddb1f8d18a3a3a07f489593a.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is differential of the proper distance.The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact. Also in constructing such equations we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.  [edit] Velocity and acceleration in 4D Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector Uμ is given by <img src="http://upload.wikimedia.org/math/0/a/a/0aad86fe0b286e12f78ac326a27a16c8.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another. Uμ also has an invariant form: <img src="http://upload.wikimedia.org/math/8/0/1/8010b2ba92ee3615ddf7d453d402108c.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. The acceleration 4-vector is given by <img src="http://upload.wikimedia.org/math/8/9/9/89919d750add5a3d3ad0d827b7fe138f.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />. Given this, differentiating the above equation by τ produces <img src="http://upload.wikimedia.org/math/7/6/3/7633590f8789139bc8d39b5bc75a56c9.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal.  [edit] Momentum in 4D The momentum and energy combine into a covariant 4-vector: <img src="http://upload.wikimedia.org/math/f/8/8/f88887b3fe8954b67947533903105c6e.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> where m is the invariant mass. The invariant magnitude of the momentum 4-vector is: <img src="http://upload.wikimedia.org/math/d/f/2/df2ea63a4adb76b34a45cde4e4b7d830.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero. <img src="http://upload.wikimedia.org/math/3/1/b/31bdac8d75746c13b1b81be222561913.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero. The rest energy is related to the mass according to the celebrated equation discussed above: <img src="http://upload.wikimedia.org/math/0/c/c/0cc50acadfdd9d19debfe42dcc154b21.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.  [edit] Force in 4D To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components. If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is: <img src="http://upload.wikimedia.org/math/b/a/d/bad3b0056a37841aae558c3fbcd4fbf8.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> where <img src="http://upload.wikimedia.org/math/d/9/5/d95fd1519e587418ebe3da8fb081701f.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is the proper time.In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing in which case it is the negative of that rate of change times c2. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e. <img src="http://upload.wikimedia.org/math/d/4/1/d41044a139defbff470177ecc63e1da5.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> while the four force is defined by the rate of change of momentum with respect to proper time, i.e. <img src="http://upload.wikimedia.org/math/3/0/0/300d8a891d489cdfa1dca05a6fdacd71.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />. In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is the negative of the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.  [edit] Relativity and unifying electromagnetism  Main article: Classical electromagnetism and special relativityTheoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation-speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.  [edit] Electromagnetism in 4D  Main article: Covariant formulation of classical electromagnetismMaxwell's equations in the 3D form are already consistent with the physical content of special relativity. But we must rewrite them to make them manifestly invariant.[22] The charge density <img src="http://upload.wikimedia.org/math/f/a/4/fa41f50d9fc4c3a61a9b6c8370a958ce.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> and current density <img src="http://upload.wikimedia.org/math/b/b/d/bbda214f893097e844ca77ead238f3ea.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> are unified into the current-charge 4-vector: <img src="http://upload.wikimedia.org/math/2/9/c/29cf0ca2d7693909b43deac80d64dcc8.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> The law of charge conservation, <img src="http://upload.wikimedia.org/math/f/a/5/fa51da272998ef8891db042f65aedb70.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />, becomes: <img src="http://upload.wikimedia.org/math/c/e/c/ceca73a06f1342b70a10f71ee190becf.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> The electric field <img src="http://upload.wikimedia.org/math/d/3/2/d32f98f806351056af76cc06b92c6675.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> and the magnetic induction <img src="http://upload.wikimedia.org/math/4/b/0/4b079b6a4f51cb469344ef7055ab9ad5.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> are now unified into the (rank 2 antisymmetric covariant) electromagnetic field tensor: <img src="http://upload.wikimedia.org/math/d/0/3/d03beb615a71022a909d6406b79c20ae.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> The density, <img src="http://upload.wikimedia.org/math/d/1/f/d1f54f559604fc13b5678b69df4afff6.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />, of the Lorentz force, <img src="http://upload.wikimedia.org/math/5/3/7/5375bcdc5fa6d4ef8618de597fad2bfd.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />, exerted on matter by the electromagnetic field becomes: <img src="http://upload.wikimedia.org/math/1/c/e/1ce6d159ad6aa075c9b67482edde3ca5.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Faraday's law of induction, <img src="http://upload.wikimedia.org/math/9/c/a/9cab6787646062d6e658cd1e83ad468f.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />, and Gauss's law for magnetism, <img src="http://upload.wikimedia.org/math/5/7/6/57619c6a86c79e56ac806faf21502c90.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />, combine to form: <img src="http://upload.wikimedia.org/math/a/6/f/a6facdd8acf16f389e85db9b290061cf.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Although there appear to be 64 equations here, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0=0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2. The electric displacement <img src="http://upload.wikimedia.org/math/c/3/e/c3e5dbe4aa735fb8c654c990852fdadf.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> and the magnetic field <img src="http://upload.wikimedia.org/math/e/4/a/e4a5c51917445501b7ac0e606c9bba14.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> are now unified into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor: <img src="http://upload.wikimedia.org/math/5/c/d/5cdebf66078fafc7602dee278423ecf2.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Ampère's law, <img src="http://upload.wikimedia.org/math/c/8/2/c8254b55c09edb6e6c394547b060efdf.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />, and Gauss's law, <img src="http://upload.wikimedia.org/math/e/b/8/eb8e03b942c5f551d3e4b2c3f1d522a4.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />, combine to form: <img src="http://upload.wikimedia.org/math/4/c/a/4ca03abedc8e5be470e22ff9476d3fd3.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> In a vacuum, the constitutive equations are: <img src="http://upload.wikimedia.org/math/8/7/4/874f2349d1a7679674f749d690ea9128.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Antisymmetry reduces these 16 equations to just six independent equations.The energy density of the electromagnetic field combines with Poynting vector and the Maxwell stress tensor to form the 4D electromagnetic stress-energy tensor. It is the flux (density) of the momentum 4-vector and as a rank 2 mixed tensor it is: <img src="http://upload.wikimedia.org/math/0/2/0/020a125b76f7ceefdc1fdfbad8407463.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> where <img src="http://upload.wikimedia.org/math/a/d/8/ad8ca1c74b2f96df1236ac875e6bd7be.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is the Kronecker delta. When upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field. The conservation of linear momentum and energy by the electromagnetic field is expressed by: <img src="http://upload.wikimedia.org/math/3/1/3/31319458233aabb429648ddf04475270.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> where <img src="http://upload.wikimedia.org/math/d/1/f/d1f54f559604fc13b5678b69df4afff6.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is again the density of the Lorentz force. This equation can be deduced from the equations above (with considerable effort).</blockquote>

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