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2012-06-14 21:58:51 UTC
The "Air shafts" of the Great Pyramid
The "Air shafts" of the Great Pyramid
http://www.ancientegyptonline.co.uk/pyramid-air-shafts.html
Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (-,+,+,+) (Some may also prefer the alternative signature (+,-,-,-); in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.) In other words, Minkowski space is a pseudo-Euclidean space with n = 4 and n - k = 1 (in a broader definition any n > 1 is allowed). Elements of Minkowski space are called events or four-vectors. Minkowski space is often denoted R1,3 to emphasize the signature, although it is also denoted M4 or simply M. It is perhaps the simplest example of a pseudo-Riemannian manifold.
[edit] The Minkowski inner product
This inner product is similar to the usual, Euclidean, inner product, but is used to describe a different geometry; the geometry is usually associated with relativity. Let M be a 4-dimensional real vector space. The Minkowski inner product is a map ?: M × M ? R (i.e. given any two vectors v, w in M we define ?(v,w) as a real number) which satisfies properties (1), (2), (3) listed here, as well as property (4) given below:
1. bilinear ?(au+v, w) = a?(u,w) + ?(v,w)
for all a ? R and u, v, w in M.
2 symmetric ?(v,w) = ?(w,v)
for all v, w ? M.
3. nondegenerate if ?(v,w) = 0 for all w ? M then v = 0.Note that this is not an inner product in the usual sense, since it is not positive-definite, i.e. the quadratic form ||v||2 = ?(v,v) need not be positive. The positive-definite condition has been replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa). The inner product is said to be indefinite. These misnomers, "Minkowski inner product" and "Minkowski metric" conflict with the standard meanings of inner product and metric in pure mathematics; as with many other misnomers the usage of these terms is due to similarity to the mathematical structure.
Just as in Euclidean space, two vectors v and w are said to be orthogonal if ?(v,w) = 0. But Minkowski space differs by including hyperbolic-orthogonal events in case v and w span a plane where ? takes negative values. This difference is clarified by comparing the Euclidean structure of the ordinary complex number plane to the structure of the plane of split-complex numbers. The Minkowski norm of a vector v is defined by
\|v\| = \sqrt{|\eta(v,v)|}.
This is not a norm in the usual sense (it fails to be subadditive), but it does define a useful generalization of the notion of length to Minkowski space. In particular, a vector v is called a unit vector if ||v|| = 1 (i.e., ?(v,v) = ±1). A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis.
By the Gram–Schmidt process, any inner product space satisfying conditions 1 to 3 above always has an orthonormal basis. Furthermore, the number of positive and negative unit vectors in any such basis is a fixed pair of numbers, equal to the signature of the inner product. This is Sylvester's law of inertia.
Then the fourth condition on ? can be stated:
4. signature The bilinear form ? has signature (-,+,+,+) or (+,-,-,-).
Which signature is used is a matter of convention. Both are fairly common. See sign convention.
[edit] Standard basis
A standard basis for Minkowski space is a set of four mutually orthogonal vectors {e0,e1,e2,e3} such that
-(e0)2 = (e1)2 = (e2)2 = (e3)2 = 1
These conditions can be written compactly in the following form:
\langle e_\mu, e_\nu \rangle = \eta_{\mu \nu}
where µ and ? run over the values (0, 1, 2, 3) and the matrix ? is given by
\eta = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}
This tensor is frequently called the "Minkowski tensor". Relative to a standard basis, the components of a vector v are written (v0,v1,v2,v3) and we use the Einstein notation to write v = vµeµ. The component v0 is called the timelike component of v while the other three components are called the spatial components.
In terms of components, the inner product between two vectors v and w is given by
\langle v, w \rangle = \eta_{\mu \nu} v^\mu w^\nu = - v^0 w^0 + v^1 w^1 + v^2 w^2 + v^3 w^3
and the norm-squared of a vector v is
v2 = ?µ? vµv? = -(v0)2 + (v1)2 + (v2)2 + (v3)2
[edit] Alternative definition
The section above defines Minkowski space as a vector space. There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the Poincaré group with the Lorentz group as the stabilizer. See Erlangen program.
Note also that the term "Minkowski space" is also used for analogues in any dimension: if n=2, n-dimensional Minkowski space is a vector space or affine space of real dimension n on which there is an inner product or pseudo-Riemannian metric of signature (n-1,1), i.e., in the above terminology, n-1 "pluses" and one "minus".
[edit] Lorentz transformations
[icon] This section requires expansion.
Further information: Lorentz transformation, Lorentz group, and Poincaré group
Standard configuration of coordinate systems for Lorentz transformations.
All four-vectors, that is, vectors in Minkowski space, transform in the same manner. In the standard sets of inertial frames as shown by the graph,
\begin{bmatrix} U'_0 \\ U'_1 \\ U'_2 \\ U'_3 \end{bmatrix} = \begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} U_0 \\ U_1 \\ U_2 \\ U_3 \end{bmatrix}\ .
where
\beta = { v \over c}
and
\gamma = { 1 \over \sqrt{1 - {v^2 \over c^2}} }
[edit] Symmetries
One of the symmetries of Minkowski space is called a "Lorentz boost". This symmetry is often illustrated with a Minkowski diagram.
The Poincaré group is the group of isometries of Minkowski spacetime.
[edit] Causal structure
Main article: Causal structure
Vectors are classified according to the sign of ?(v,v). When the standard signature (-,+,+,+) is used, a vector v is:
Timelike if ?(v,v) < 0
Spacelike if ?(v,v) > 0
Null (or lightlike) if ?(v,v) = 0
This terminology comes from the use of Minkowski space in the theory of relativity. The set of all null vectors at an event of Minkowski space constitutes the light cone of that event. Note that all these notions are independent of the frame of reference. Given a timelike vector v, there is a worldline of constant velocity associated with it. The set {w : ?(w,v) = 0 } corresponds to the simultaneous hyperplane at the origin of this worldline. Minkowski space exhibits relativity of simultaneity since this hyperplane depends on v. In the plane spanned by v and such a w in the hyperplane, the relation of w to v is hyperbolic-orthogonal.
Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors we have
future directed timelike vectors whose first component is positive, and
past directed timelike vectors whose first component is negative.
Null vectors fall into three classes:
the zero vector, whose components in any basis are (0,0,0,0),
future directed null vectors whose first component is positive, and
past directed null vectors whose first component is negative.
Together with spacelike vectors there are 6 classes in all.
An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis. Over the reals, if two null vectors are orthogonal (zero inner product), then they must be proportional. However, allowing complex numbers, one can obtain a null tetrad which is a basis consisting of null vectors, some of which are orthogonal to each other.
Vector fields are called timelike, spacelike or null if the associated vectors are timelike, spacelike or null at each point where the field is defined.
[edit] Causality relations
Let x, y ? M. We say that
x chronologically precedes y if y - x is future directed timelike.
x causally precedes y if y - x is future directed null
[edit] Reversed triangle inequality
If v and w are two equally directed timelike four-vectors, then
|v+w| \ge |v|+|w|,
where
|v|:=\sqrt{-\eta_{\mu \nu}v^\mu v^\nu}.
[edit] Locally flat spacetime
Strictly speaking, the use of the Minkowski space to describe physical systems over finite distances applies only in the Newtonian limit of systems without significant gravitation. In the case of significant gravitation, spacetime becomes curved and one must abandon special relativity in favor of the full theory of general relativity.
Nevertheless, even in such cases, Minkowski space is still a good description in an infinitesimally small region surrounding any point (barring gravitational singularities). More abstractly, we say that in the presence of gravity spacetime is described by a curved 4-dimensional manifold for which the tangent space to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.
In the realm of weak gravity, spacetime becomes flat and looks globally, not just locally, like Minkowski space. For this reason Minkowski space is often referred to as flat spacetime.Vectors
Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example a general point might have position vector a, equal to
\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.
This can be written in terms of the four standard basis vectors (e1, e2, e3, e4), given by
\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}; \mathbf{e}_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix},
so the general vector a is
\mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.
Vectors add, subtract and scale as in three dimensions.
The dot product of Euclidean three-dimensional space generalizes to four dimensions as
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.
It can be used to calculate the norm or length of a vector,
\left| \mathbf{a} \right| = \sqrt{\mathbf{a} \cdot \mathbf{a} } = \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2},
and calculate or define the angle between two vectors as
\theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.
Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate pairing different from the dot product:
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4.
As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing b_4 actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.
The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:
\begin{align} \mathbf{a} \wedge \mathbf{b} = (a_1b_2 - a_2b_1)\mathbf{e}_{12} + (a_1b_3 - a_3b_1)\mathbf{e}_{13} + (a_1b_4 - a_4b_1)\mathbf{e}_{14} + (a_2b_3 - a_3b_2)\mathbf{e}_{23} \\ + (a_2b_4 - a_4b_2)\mathbf{e}_{24} + (a_3b_4 - a_4b_3)\mathbf{e}_{34}. \end{align}
This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e12, e13, e14, e23, e24, e34). They can be used to generate rotations in four dimensions.
[edit] Orthogonality and vocabulary
In the familiar 3-dimensional space that we live in there are three coordinate axes — usually labeled x, y, and z — with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude. Lengths measured along these axes can be called height, width, and depth.
Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively. A length measured along the w axis can be called spissitude, as coined by Henry More.Theoretical particle physics attempts to develop the models, theoretical framework, and mathematical tools to understand current experiments and make predictions for future experiments. See also theoretical physics. There are several major interrelated efforts in theoretical particle physics today. One important branch attempts to better understand the Standard Model and its tests. By extracting the parameters of the Standard Model from experiments with less uncertainty, this work probes the limits of the Standard Model and therefore expands our understanding of nature's building blocks. These efforts are made challenging by the difficulty of calculating quantities in quantum chromodynamics. Some theorists working in this area refer to themselves as phenomenologists and may use the tools of quantum field theory and effective field theory. Others make use of lattice field theory and call themselves lattice theorists.
Another major effort is in model building where model builders develop ideas for what physics may lie beyond the Standard Model (at higher energies or smaller distances). This work is often motivated by the hierarchy problem and is constrained by existing experimental data. It may involve work on supersymmetry, alternatives to the Higgs mechanism, extra spatial dimensions (such as the Randall-Sundrum models), Preon theory, combinations of these, or other ideas.
A third major effort in theoretical particle physics is string theory. String theorists attempt to construct a unified description of quantum mechanics and general relativity by building a theory based on small strings, and branes rather than particles. If the theory is successful, it may be considered a "Theory of Everything".
There are also other areas of work in theoretical particle physics ranging from particle cosmology to loop quantum gravity.
This division of efforts in particle physics is reflected in the names of categories on the arXiv, a preprint archive [1]: hep-th (theory), hep-ph (phenomenology), hep-ex (experiments), hep-lat (lattice gauge theory).Resonance (particle physics)
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The ?(1S) resonance, as observed by the E288 collaboration, headed by Leon Lederman, at Fermilab in 1977. The resonance is located at 9.5 GeV, corresponding to the mass of the ?(1S).
In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. These peaks are associated with subatomic particles (such as nucleons, delta baryons, upsilon mesons) and their excitations. The width of the resonance (G) is related to the lifetime (t) of the particle (or its excited state) by the relation
\Gamma=\frac{\hbar}{\tau}
where h is the reduced planck constant.Magnetic monopole
From Wikipedia, the free encyclopedia
(Redirected from Magnetic Monopole)
Jump to: navigation, search
It is impossible to make magnetic monopoles from a bar magnet. If a bar magnet is cut in half, it is not the case that one half has the north pole and the other half has the south pole. Instead, each piece has its own north and south poles. A magnetic monopole cannot be created from normal matter such as atoms and electrons, but would instead be a new elementary particle.
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole (a north pole without a south pole or vice-versa).[1][2] In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence.[3][4] Magnetism in bar magnets and electromagnets does not arise from magnetic monopoles, and in fact there is no conclusive experimental evidence that magnetic monopoles exist at all in the universe.http://en.wikipedia.org/wiki/Magnetic_MonopoleMagnets exert forces on one another, similar to the force associated with electric charges. Like poles will repel each other, and unlike poles will attract. When any magnet (an object conventionally described as having magnetic north and south poles) is cut in half across the axis joining those "poles", the resulting pieces are two normal (albeit smaller) magnets. Each has its own north pole and south pole.
Even atoms and subatomic particles have tiny magnetic fields. In the Bohr model of an atom, electrons orbit the nucleus. Their constant motion gives rise to a magnetic field. Permanent magnets have measurable magnetic fields because the atoms and molecules in them are arranged in such a way that their individual magnetic fields align, combining to form large aggregate fields. In this model, the lack of a single pole makes intuitive sense: cutting a bar magnet in half does nothing to the arrangement of the molecules within. The end result is two bar magnetics whose atoms have the same orientation as before, and therefore generate a magnetic field with the same orientation as the original larger magnet.Maxwell's equations of electromagnetism relate the electric and magnetic fields to each other and to the motions of electric charges. The standard equations provide for electric charges, but they posit no magnetic charges. Except for this difference, the equations are symmetric under the interchange of the electric and magnetic fields.[15] In fact, symmetric Maxwell's equations can be written when all charges (and hence electric currents) are zero, and this is how the electromagnetic wave equation is derived.
Fully symmetric Maxwell's equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.[16] With the inclusion of a variable for the density of these magnetic charges, say ?m, there will also be a "magnetic current density" variable in the equations, jm.
If magnetic charges do not exist - or if they do exist but are not present in a region of space - then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as ?·B = 0 (where ? is divergence and B is the magnetic B field).
For a long time, the open question has been "Why does the magnetic charge always seem to be zero?"Dirac string
Main article: Dirac string
A gauge theory like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.
In electrodynamics, the group is U(1), unit complex numbers under multiplication. For infinitesimal paths, the group element is 1+iAµdxµ which implies that for finite paths parametrized by s, the group element is:
\prod_s \left( 1+ieA_\mu {dx^\mu \over ds} ds \right) = \exp \left( ie\int A\cdot dx \right) .
The map from paths to group elements is called the Wilson loop or the holonomy, and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:
e \oint_{\partial D} A\cdot dx = e \int_D (\nabla \times A) dS = e \int_D B dS.
So that the phase a charged particle gets when going in a loop is the magnetic flux through the loop. When a small solenoid has a magnetic flux, there are interference fringes for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.
But if all particle charges are integer multiples of e, solenoids with a flux of 2p/e have no interference fringes, because the phase factor for any charged particle is e2pi = 1. Such a solenoid, if thin enough, is quantum-mechanically invisible. If such a solenoid were to carry a flux of 2p/e, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.
Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.String theory
In our universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by Hawking radiation, the lightest charged particles cannot be too heavy. The lightest monopole should have a mass less than or comparable to its charge in natural units.
So in a consistent holographic theory, of which string theory is the only known example, there are always finite-mass monopoles. For ordinary electromagnetism, the mass bound is not very useful because it is about same size as the Planck mass.The "Air shafts" of the Great Pyramid
The "Air shafts" of the Great Pyramid
http://www.ancientegyptonline.co.uk/pyramid-air-shafts.html
Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (-,+,+,+) (Some may also prefer the alternative signature (+,-,-,-); in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.) In other words, Minkowski space is a pseudo-Euclidean space with n = 4 and n - k = 1 (in a broader definition any n > 1 is allowed). Elements of Minkowski space are called events or four-vectors. Minkowski space is often denoted R1,3 to emphasize the signature, although it is also denoted M4 or simply M. It is perhaps the simplest example of a pseudo-Riemannian manifold.
[edit] The Minkowski inner product
This inner product is similar to the usual, Euclidean, inner product, but is used to describe a different geometry; the geometry is usually associated with relativity. Let M be a 4-dimensional real vector space. The Minkowski inner product is a map ?: M × M ? R (i.e. given any two vectors v, w in M we define ?(v,w) as a real number) which satisfies properties (1), (2), (3) listed here, as well as property (4) given below:
1. bilinear ?(au+v, w) = a?(u,w) + ?(v,w)
for all a ? R and u, v, w in M.
2 symmetric ?(v,w) = ?(w,v)
for all v, w ? M.
3. nondegenerate if ?(v,w) = 0 for all w ? M then v = 0.Note that this is not an inner product in the usual sense, since it is not positive-definite, i.e. the quadratic form ||v||2 = ?(v,v) need not be positive. The positive-definite condition has been replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa). The inner product is said to be indefinite. These misnomers, "Minkowski inner product" and "Minkowski metric" conflict with the standard meanings of inner product and metric in pure mathematics; as with many other misnomers the usage of these terms is due to similarity to the mathematical structure.
Just as in Euclidean space, two vectors v and w are said to be orthogonal if ?(v,w) = 0. But Minkowski space differs by including hyperbolic-orthogonal events in case v and w span a plane where ? takes negative values. This difference is clarified by comparing the Euclidean structure of the ordinary complex number plane to the structure of the plane of split-complex numbers. The Minkowski norm of a vector v is defined by
\|v\| = \sqrt{|\eta(v,v)|}.
This is not a norm in the usual sense (it fails to be subadditive), but it does define a useful generalization of the notion of length to Minkowski space. In particular, a vector v is called a unit vector if ||v|| = 1 (i.e., ?(v,v) = ±1). A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis.
By the Gram–Schmidt process, any inner product space satisfying conditions 1 to 3 above always has an orthonormal basis. Furthermore, the number of positive and negative unit vectors in any such basis is a fixed pair of numbers, equal to the signature of the inner product. This is Sylvester's law of inertia.
Then the fourth condition on ? can be stated:
4. signature The bilinear form ? has signature (-,+,+,+) or (+,-,-,-).
Which signature is used is a matter of convention. Both are fairly common. See sign convention.
[edit] Standard basis
A standard basis for Minkowski space is a set of four mutually orthogonal vectors {e0,e1,e2,e3} such that
-(e0)2 = (e1)2 = (e2)2 = (e3)2 = 1
These conditions can be written compactly in the following form:
\langle e_\mu, e_\nu \rangle = \eta_{\mu \nu}
where µ and ? run over the values (0, 1, 2, 3) and the matrix ? is given by
\eta = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}
This tensor is frequently called the "Minkowski tensor". Relative to a standard basis, the components of a vector v are written (v0,v1,v2,v3) and we use the Einstein notation to write v = vµeµ. The component v0 is called the timelike component of v while the other three components are called the spatial components.
In terms of components, the inner product between two vectors v and w is given by
\langle v, w \rangle = \eta_{\mu \nu} v^\mu w^\nu = - v^0 w^0 + v^1 w^1 + v^2 w^2 + v^3 w^3
and the norm-squared of a vector v is
v2 = ?µ? vµv? = -(v0)2 + (v1)2 + (v2)2 + (v3)2
[edit] Alternative definition
The section above defines Minkowski space as a vector space. There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the Poincaré group with the Lorentz group as the stabilizer. See Erlangen program.
Note also that the term "Minkowski space" is also used for analogues in any dimension: if n=2, n-dimensional Minkowski space is a vector space or affine space of real dimension n on which there is an inner product or pseudo-Riemannian metric of signature (n-1,1), i.e., in the above terminology, n-1 "pluses" and one "minus".
[edit] Lorentz transformations
[icon] This section requires expansion.
Further information: Lorentz transformation, Lorentz group, and Poincaré group
Standard configuration of coordinate systems for Lorentz transformations.
All four-vectors, that is, vectors in Minkowski space, transform in the same manner. In the standard sets of inertial frames as shown by the graph,
\begin{bmatrix} U'_0 \\ U'_1 \\ U'_2 \\ U'_3 \end{bmatrix} = \begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} U_0 \\ U_1 \\ U_2 \\ U_3 \end{bmatrix}\ .
where
\beta = { v \over c}
and
\gamma = { 1 \over \sqrt{1 - {v^2 \over c^2}} }
[edit] Symmetries
One of the symmetries of Minkowski space is called a "Lorentz boost". This symmetry is often illustrated with a Minkowski diagram.
The Poincaré group is the group of isometries of Minkowski spacetime.
[edit] Causal structure
Main article: Causal structure
Vectors are classified according to the sign of ?(v,v). When the standard signature (-,+,+,+) is used, a vector v is:
Timelike if ?(v,v) < 0
Spacelike if ?(v,v) > 0
Null (or lightlike) if ?(v,v) = 0
This terminology comes from the use of Minkowski space in the theory of relativity. The set of all null vectors at an event of Minkowski space constitutes the light cone of that event. Note that all these notions are independent of the frame of reference. Given a timelike vector v, there is a worldline of constant velocity associated with it. The set {w : ?(w,v) = 0 } corresponds to the simultaneous hyperplane at the origin of this worldline. Minkowski space exhibits relativity of simultaneity since this hyperplane depends on v. In the plane spanned by v and such a w in the hyperplane, the relation of w to v is hyperbolic-orthogonal.
Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors we have
future directed timelike vectors whose first component is positive, and
past directed timelike vectors whose first component is negative.
Null vectors fall into three classes:
the zero vector, whose components in any basis are (0,0,0,0),
future directed null vectors whose first component is positive, and
past directed null vectors whose first component is negative.
Together with spacelike vectors there are 6 classes in all.
An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis. Over the reals, if two null vectors are orthogonal (zero inner product), then they must be proportional. However, allowing complex numbers, one can obtain a null tetrad which is a basis consisting of null vectors, some of which are orthogonal to each other.
Vector fields are called timelike, spacelike or null if the associated vectors are timelike, spacelike or null at each point where the field is defined.
[edit] Causality relations
Let x, y ? M. We say that
x chronologically precedes y if y - x is future directed timelike.
x causally precedes y if y - x is future directed null
[edit] Reversed triangle inequality
If v and w are two equally directed timelike four-vectors, then
|v+w| \ge |v|+|w|,
where
|v|:=\sqrt{-\eta_{\mu \nu}v^\mu v^\nu}.
[edit] Locally flat spacetime
Strictly speaking, the use of the Minkowski space to describe physical systems over finite distances applies only in the Newtonian limit of systems without significant gravitation. In the case of significant gravitation, spacetime becomes curved and one must abandon special relativity in favor of the full theory of general relativity.
Nevertheless, even in such cases, Minkowski space is still a good description in an infinitesimally small region surrounding any point (barring gravitational singularities). More abstractly, we say that in the presence of gravity spacetime is described by a curved 4-dimensional manifold for which the tangent space to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.
In the realm of weak gravity, spacetime becomes flat and looks globally, not just locally, like Minkowski space. For this reason Minkowski space is often referred to as flat spacetime.Vectors
Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example a general point might have position vector a, equal to
\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.
This can be written in terms of the four standard basis vectors (e1, e2, e3, e4), given by
\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}; \mathbf{e}_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix},
so the general vector a is
\mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.
Vectors add, subtract and scale as in three dimensions.
The dot product of Euclidean three-dimensional space generalizes to four dimensions as
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.
It can be used to calculate the norm or length of a vector,
\left| \mathbf{a} \right| = \sqrt{\mathbf{a} \cdot \mathbf{a} } = \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2},
and calculate or define the angle between two vectors as
\theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.
Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate pairing different from the dot product:
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4.
As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing b_4 actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.
The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:
\begin{align} \mathbf{a} \wedge \mathbf{b} = (a_1b_2 - a_2b_1)\mathbf{e}_{12} + (a_1b_3 - a_3b_1)\mathbf{e}_{13} + (a_1b_4 - a_4b_1)\mathbf{e}_{14} + (a_2b_3 - a_3b_2)\mathbf{e}_{23} \\ + (a_2b_4 - a_4b_2)\mathbf{e}_{24} + (a_3b_4 - a_4b_3)\mathbf{e}_{34}. \end{align}
This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e12, e13, e14, e23, e24, e34). They can be used to generate rotations in four dimensions.
[edit] Orthogonality and vocabulary
In the familiar 3-dimensional space that we live in there are three coordinate axes — usually labeled x, y, and z — with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude. Lengths measured along these axes can be called height, width, and depth.
Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively. A length measured along the w axis can be called spissitude, as coined by Henry More.Theoretical particle physics attempts to develop the models, theoretical framework, and mathematical tools to understand current experiments and make predictions for future experiments. See also theoretical physics. There are several major interrelated efforts in theoretical particle physics today. One important branch attempts to better understand the Standard Model and its tests. By extracting the parameters of the Standard Model from experiments with less uncertainty, this work probes the limits of the Standard Model and therefore expands our understanding of nature's building blocks. These efforts are made challenging by the difficulty of calculating quantities in quantum chromodynamics. Some theorists working in this area refer to themselves as phenomenologists and may use the tools of quantum field theory and effective field theory. Others make use of lattice field theory and call themselves lattice theorists.
Another major effort is in model building where model builders develop ideas for what physics may lie beyond the Standard Model (at higher energies or smaller distances). This work is often motivated by the hierarchy problem and is constrained by existing experimental data. It may involve work on supersymmetry, alternatives to the Higgs mechanism, extra spatial dimensions (such as the Randall-Sundrum models), Preon theory, combinations of these, or other ideas.
A third major effort in theoretical particle physics is string theory. String theorists attempt to construct a unified description of quantum mechanics and general relativity by building a theory based on small strings, and branes rather than particles. If the theory is successful, it may be considered a "Theory of Everything".
There are also other areas of work in theoretical particle physics ranging from particle cosmology to loop quantum gravity.
This division of efforts in particle physics is reflected in the names of categories on the arXiv, a preprint archive [1]: hep-th (theory), hep-ph (phenomenology), hep-ex (experiments), hep-lat (lattice gauge theory).Resonance (particle physics)
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The ?(1S) resonance, as observed by the E288 collaboration, headed by Leon Lederman, at Fermilab in 1977. The resonance is located at 9.5 GeV, corresponding to the mass of the ?(1S).
In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. These peaks are associated with subatomic particles (such as nucleons, delta baryons, upsilon mesons) and their excitations. The width of the resonance (G) is related to the lifetime (t) of the particle (or its excited state) by the relation
\Gamma=\frac{\hbar}{\tau}
where h is the reduced planck constant.Magnetic monopole
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It is impossible to make magnetic monopoles from a bar magnet. If a bar magnet is cut in half, it is not the case that one half has the north pole and the other half has the south pole. Instead, each piece has its own north and south poles. A magnetic monopole cannot be created from normal matter such as atoms and electrons, but would instead be a new elementary particle.
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole (a north pole without a south pole or vice-versa).[1][2] In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence.[3][4] Magnetism in bar magnets and electromagnets does not arise from magnetic monopoles, and in fact there is no conclusive experimental evidence that magnetic monopoles exist at all in the universe.http://en.wikipedia.org/wiki/Magnetic_MonopoleMagnets exert forces on one another, similar to the force associated with electric charges. Like poles will repel each other, and unlike poles will attract. When any magnet (an object conventionally described as having magnetic north and south poles) is cut in half across the axis joining those "poles", the resulting pieces are two normal (albeit smaller) magnets. Each has its own north pole and south pole.
Even atoms and subatomic particles have tiny magnetic fields. In the Bohr model of an atom, electrons orbit the nucleus. Their constant motion gives rise to a magnetic field. Permanent magnets have measurable magnetic fields because the atoms and molecules in them are arranged in such a way that their individual magnetic fields align, combining to form large aggregate fields. In this model, the lack of a single pole makes intuitive sense: cutting a bar magnet in half does nothing to the arrangement of the molecules within. The end result is two bar magnetics whose atoms have the same orientation as before, and therefore generate a magnetic field with the same orientation as the original larger magnet.Maxwell's equations of electromagnetism relate the electric and magnetic fields to each other and to the motions of electric charges. The standard equations provide for electric charges, but they posit no magnetic charges. Except for this difference, the equations are symmetric under the interchange of the electric and magnetic fields.[15] In fact, symmetric Maxwell's equations can be written when all charges (and hence electric currents) are zero, and this is how the electromagnetic wave equation is derived.
Fully symmetric Maxwell's equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.[16] With the inclusion of a variable for the density of these magnetic charges, say ?m, there will also be a "magnetic current density" variable in the equations, jm.
If magnetic charges do not exist - or if they do exist but are not present in a region of space - then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as ?·B = 0 (where ? is divergence and B is the magnetic B field).
For a long time, the open question has been "Why does the magnetic charge always seem to be zero?"Dirac string
Main article: Dirac string
A gauge theory like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.
In electrodynamics, the group is U(1), unit complex numbers under multiplication. For infinitesimal paths, the group element is 1+iAµdxµ which implies that for finite paths parametrized by s, the group element is:
\prod_s \left( 1+ieA_\mu {dx^\mu \over ds} ds \right) = \exp \left( ie\int A\cdot dx \right) .
The map from paths to group elements is called the Wilson loop or the holonomy, and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:
e \oint_{\partial D} A\cdot dx = e \int_D (\nabla \times A) dS = e \int_D B dS.
So that the phase a charged particle gets when going in a loop is the magnetic flux through the loop. When a small solenoid has a magnetic flux, there are interference fringes for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.
But if all particle charges are integer multiples of e, solenoids with a flux of 2p/e have no interference fringes, because the phase factor for any charged particle is e2pi = 1. Such a solenoid, if thin enough, is quantum-mechanically invisible. If such a solenoid were to carry a flux of 2p/e, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.
Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.String theory
In our universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by Hawking radiation, the lightest charged particles cannot be too heavy. The lightest monopole should have a mass less than or comparable to its charge in natural units.
So in a consistent holographic theory, of which string theory is the only known example, there are always finite-mass monopoles. For ordinary electromagnetism, the mass bound is not very useful because it is about same size as the Planck mass.The "Air shafts" of the Great Pyramid