Additions:
It follows immediately that, using unprimed Minkowski coordinates, switching ""m"" and ""n"" makes no difference to ""
"" Hence, in any coordinates, ""gi'j' = gj'i'"". In other words, ""g"" is symmetrical under interchange of indices.
Deletions:
It follows immediately that using unprimed Minkowski coordinates, switching ""m"" and ""n"" makes no difference to """". Hence, in any coordinates, ""gi'j' = gj'i'"". In other words, ""g"" is symmetrical under interchange of indices.
Additions:
It follows immediately that using unprimed Minkowski coordinates, switching ""m"" and ""n"" makes no difference to """". Hence, in any coordinates, ""gi'j' = gj'i'"". In other words, ""g"" is symmetrical under interchange of indices.
Deletions:
It follows immediately that using unprimed Minkowski coordinates, switching ""m"" and ""n"" makes no difference to """", and hence that in any coordinates ""gi'j' = gj'i'"". In other words, ""g"" is symmetrical under interchange of indices.
Additions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1"", ""g11 = g22 = g33 = −1"", and all other entries equal to zero. In ""spherical coordinates"" Minkowski spacetime has metric with ""g00 = 1"", ""g11 = −1"", ""g22 = −r2"", ""g33 = −r2sin2θ"" and all other entries equal to zero.
It follows immediately that using unprimed Minkowski coordinates, switching ""m"" and ""n"" makes no difference to """", and hence that in any coordinates ""gi'j' = gj'i'"". In other words, ""g"" is symmetrical under interchange of indices.
Deletions:
Additions:
Deletions:
Deletions:
[[http://en.wikipedia.org/wiki/Tensor Tensors]] are built from vectors. They provide the mathematical structure used to describe states of many particles in quantum mechanics, as well as the structure to express general physical law in general relativity. The idea in their construction is remarkably straightforward. Tensors simplify physical laws and show that ""
""====""""====
[[IntroductionToTensors IntroductionToTensors ↑]] [[Gravity Gravity →]]the magnetic force is just the electrostatic force after Lorentz transformation.
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1"", ""g11 = g22 = g22 = −1"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g<sub>00"", ""g11 = −1"", ""g22 = −r^2"", ""g22 = −r^2sin2;theta;"" and all other entries equal to zero.
Additions:
[[http://en.wikipedia.org/wiki/Tensor Tensors]] are built from vectors. They provide the mathematical structure used to describe states of many particles in quantum mechanics, as well as the structure to express general physical law in general relativity. The idea in their construction is remarkably straightforward. Tensors simplify physical laws and show that """"====""""====
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1"", ""g11 = g22 = g33 = −1"", and all other entries equal to zero. In ""spherical coordinates"" Minkowski spacetime has metric with ""gsub>00"", ""g11 = −1"", ""g22 = −r2"", ""g33 = −r2sin2θ"" and all other entries equal to zero.
[[IntroductionToTensors IntroductionToTensors ↑]] [[Gravity Gravity →]]the magnetic force is just the electrostatic force after Lorentz transformation.
Additions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1"", ""g11 = g22 = g22 = −1"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g<sub>00"", ""g11 = −1"", ""g22 = −r^2"", ""g22 = −r^2sin2;theta;"" and all other entries equal to zero.
Deletions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1"", ""g11 = g22 = g22 = −1"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1"", ""g11 = −1"", ""g22 = −r^2"", ""g22 = −r^2sin2;theta;"" and all other entries equal to zero.
Additions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1"", ""g11 = g22 = g22 = −1"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1"", ""g11 = −1"", ""g22 = −r^2"", ""g22 = −r^2sin2;theta;"" and all other entries equal to zero.
Deletions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
Additions:
<<""Definition: A tensor space is a vector space built from other vector spaces using three simple rules:""
""Rule 1. If xi and y j are vectors, then the object consisting of the products of the coefficients, xiy j, is a tensor.""
""Rule 2. If xij and yij are tensors and a and b are scalars, then axij + byij is a tensor.""
""Rule 3. Addition and multiplication by scalars obey the rules of vector space.""
""""{{image class="right" alt="Tensors-7" title="Basis tensors" url="images/tensors/Tensors-7N.gif"}}For any basis vector, ""
"", in an ""n""-dimensional vector space, ""V1"", and any basis vector, ""
"", in an ""m""-dimensional vector space, ""V2"", make a new object ""
"" (""V1"", and ""V2"" may be the same vector space). The ""mn"" objects of the form """" can be represented as ""mn × 1"" column vectors, each with a ""1"" in one position and a zero in all others (this is how they are represented in computer memory). Introduce the usual rules of matrix addition and multiplication by scalars. The ""mn""-dimensional vector space built like this is a tensor space. Call this tensor space ""T"". The objects in ""T"" are tensors. The objects ""
"" are a basis for ""T"". We write ""
"".
<<""Definition: 
is the tensor (or outer) product.""
""
""
in ""V1"" and
""
""
in ""V2"", there is an object in ""T"":
""
""
Rank 2 tensors are formed from the product of two vector spaces, and have coefficents with two vector indices. One may find the tensor product of any number of vector spaces in exactly the same way. Rank ""n"" tensors have coefficients with ""n"" indices.
<<""Definition: The tensor product of n vector spaces, V2, … Vn"",
""is a rank n tensor space.""
Tensors are built from vectors, and vector properties apply to each index of a tensor individually. We can apply the index lowering operator, or ""metric"", ""gij"" to change a contravariant index to a covariant one, and the raising operator ""gij"" to restore it to contravariance, e.g. for a Rank 3 tensor, ""Aijk"", we have,
Other indexed quantities appear, known as //non-tensors//. One can raise and lower the indices of non-tensors, using ""g"", but non-tensors are distinguished from tensors because their indices do not satisfy vector transformation laws (any indexed quantity satisfying vector transformation laws can be written as a sum of basis elements and is therefore a tensor). Some treatments even “define” tensors by their transformation properties. In my view, that obscures the mathematical structure of a tensor space.
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor.
//Proof//: since ""xiAijk"" is a tensor
This holds for all ""xi"". So,
So, ""Aijk"" obeys the tensor transformation law. The quotient theorem holds if ""Aijk"" is replaced with a quantity with any number of upstairs or downstairs indices.
To preserve the scalar product under general coordinate transformation we require that, for any vectors ""x"" and ""y"",
Since this is true for all values of ""xi'"" and ""y j'"", we have
Thus, the transformation law for ""gij"" is as for covariant vectors for each suffix. Similarly, the transformation law for ""gij"" is as for contravariant vectors for each superfix. Thus, the ""metric"", ""g"", is a tensor, known as the //metric tensor//.
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
A lower rank tensor is sometimes created from a higher one by //contraction//. This means setting upper and lower indices equal and using the summation convention. For example, we might wish to define a vector, ""Ai"", from the rank 3 tensor ""Aijk"" by contracting the second and third indices thus, ""Ai = Aijj"" .We also talk of contracting an index of one tensor quantity with an index of another. The contraction of the second index of ""Bij"" with the first of ""Cij"" is a rank 2 tensor, ""BijC jk"".
The //symmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally symmetric// if it is equal to its symmetric part.
The //antisymmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally antisymmetric// if it is equal to its antisymmetric part. A tensor is the sum of its symmetric and antisymmetric parts. It is easily shown that symmetry and antisymmetry are preserved by coordinate transformations — if a tensor is (anti)symmetric in one frame, it is (anti)symmetric in any frame.
The Levi-Civita tensor is defined in ""Minkowski Coordinates"" to be a totally anti-symmetric tensor, with ""ε0123 = 1"". Because of antisymmetry, ""εijkl = 0"" whenever any two indices are the same, and ""εijkl = ±1"" whenever all the indices are different. We have
>>""Definition: A field is a function on a coordinate system or a manifold."">> A tensor valued function on a ""manifold"" is a [[http://en.wikipedia.org/wiki/Tensor_field tensor field]]. A vector field is a rank 1 tensor field. The value of a tensor field at a particular point is a member of a vector or tensor space defined at that point. If we add components of a tensor defined at one point with those defined at another, we are adding members of different tensor spaces. The result is not a tensor. In particular, the ""partial derivative"" of a tensor field, ""x → f(x)"",
for a small displacement ""dxi"" in the ""i""-direction, involves the subtraction of a tensor, ""f(x)"", defined at ""x"", from one, ""f(x + dxi)"", defined at ""x + dxi"". The partial derivative of a tensor field is not, in general, a tensor.
>>""Definition: An equation is covariant if it has the same form in any coordinates, up to the naming of coordinate axes."">>The ""general principle of relativity"" states //Local laws of physics are the same irrespective of the coordinate system used to quantify them.// Vectors are not invariant, as their coordinate representation changes with the coordinate system. Relationships between vectors, defined from the dot product, are unchanged by coordinate transformation. Such relationships are said to be //covariant//. Similarly, relationships between tensors are covariant. The form of the general principle of relativity most directly applicable to classical physics is the principle of general covariance, //The equations of physics have tensorial form//.
To apply the principle of ""general covariance"" in the formulation of physical law we have to find vector and tensor quantities suitable for the description of physics. The 4-vector quantity describing a charged particle is current, ""J"". ""J"" contains information about both-the charge and the motion of the particle. For a stationary charge the 3-current is zero, so the 4-current is ""(q, 0)"". The general form of ""J"" is found by ""Lorentz transformation"" and has the form ""J = (γq, γqv)"", where ""γ"" is the ""Lorentz factor"".
>>""Definition: The electromagnetic field is described by the Faraday tensor, F."">>To describe the electromagnetic force acting on a charged particle we need to contract ""J"" with a tensor representing the electromagnetic field. This tensor is called [[http://en.wikipedia.org/wiki/Electromagnetic_tensor Faraday]], ""F"". The result of contracting ""J"" with ""F"" is a vector, force. So, Faraday is a rank 2 tensor. Faraday should express both the force acting on a charged particle, and the equal and opposite reactive force exerted by the particle on its environment. If Faraday is an antisymmetric tensor, contracting with one index will give the force on the particle and contracting with the other will give the reactive force. We write down the 4-vector law of force:
The simplest situation is a static electric field. In this case the 3-vector force is ""qE = (qEx, qEy, qEz)"". We can then write the Faraday tensor, using antisymmetry to determine the other components.
The Lorentz transformation for a ""boost"" in the ""1""-direction (i.e the ""x""-direction),
Contracting with a 4-current ""( Jt, Jx, Jy, Jz)"" gives a relativistic adjustment to the electrical force acting on the charge ""Jt"", and also introduces a force in the ""x""-direction (the direction opposite to the boost) with terms proportional to ""Jy"" and ""Jz"", i.e. perpendicular to the components of current in those directions. We identify this with magnetic force.
In the general case, Faraday is not represented simply through boosting a static field, but it is the result of fields generated by many particles, each one of which could be regarded as static in the rest frame of that particle. This would lead to a complicated expression, and we write Faraday using the resultant electric and magnetic fields, ""E = (Ex, Ey, Ez)"" and ""B = (Bx, By, Bz)"",
Deletions:
<<""Definition: A tensor space is a vector space built from other vector spaces using three simple rules:""
""Rule 1. If xi and y j are vectors, then the object consisting of the products of the coefficients, xiy j, is a tensor.""
""Rule 2. If xij and yij are tensors and a and b are scalars, then axij + byij is a tensor.""
""Rule 3. Addition and multiplication by scalars obey the rules of vector space.""
""""{{image class="right" alt="Tensors-7" title="Basis tensors" url="images/tensors/Tensors-7N.gif"}}For any basis vector, """", in an ""n""-dimensional vector space, ""V1"", and any basis vector, """", in an ""m""-dimensional vector space, ""V2"", make a new object """" (""V1"", and ""V2"" may be the same vector space). The ""mn"" objects of the form """" can be represented as ""mn × 1"" column vectors, each with a ""1"" in one position and a zero in all others (this is how they are represented in computer memory). Introduce the usual rules of matrix addition and multiplication by scalars. The ""mn""-dimensional vector space built like this is a tensor space. Call this tensor space ""T"". The objects in ""T"" are tensors. The objects """" are a basis for ""T"". We write """".
<<""Definition: is the tensor (or outer) product.""
""""
in ""V1"" and
""""
in ""V2"", there is an object in ""T"":
""""
Rank 2 tensors are formed from the product of two vector spaces, and have coefficents with two vector indices. One may find the tensor product of any number of vector spaces in exactly the same way. Rank ""n"" tensors have coefficients with ""n"" indices.
<<""Definition: The tensor product of n vector spaces, V2, … Vn"",
""is a rank n tensor space.""
Tensors are built from vectors, and vector properties apply to each index of a tensor individually. We can apply the index lowering operator, or ""metric"", ""gij"" to change a contravariant index to a covariant one, and the raising operator ""gij"" to restore it to contravariance, e.g. for a Rank 3 tensor, ""Aijk"", we have,
Other indexed quantities appear, known as //non-tensors//. One can raise and lower the indices of non-tensors, using ""g"", but non-tensors are distinguished from tensors because their indices do not satisfy vector transformation laws (any indexed quantity satisfying vector transformation laws can be written as a sum of basis elements and is therefore a tensor). Some treatments even “define” tensors by their transformation properties. In my view, that obscures the mathematical structure of a tensor space.
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor.
//Proof//: since ""xiAijk"" is a tensor
This holds for all ""xi"". So,
So, ""Aijk"" obeys the tensor transformation law. The quotient theorem holds if ""Aijk"" is replaced with a quantity with any number of upstairs or downstairs indices.
To preserve the scalar product under general coordinate transformation we require that, for any vectors ""x"" and ""y"",
Since this is true for all values of ""xi'"" and ""y j'"", we have
Thus, the transformation law for ""gij"" is as for covariant vectors for each suffix. Similarly, the transformation law for ""gij"" is as for contravariant vectors for each superfix. Thus, the ""metric"", ""g"", is a tensor, known as the //metric tensor//.
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\s/span>"", ""g11 = g22 = g22 = −1<\s/span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\s/span>"", ""g11 = −1<\s/span>"", ""g22 = −r^2<\sup><\s/span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\s/span>"" and all other entries equal to zero.
A lower rank tensor is sometimes created from a higher one by //contraction//. This means setting upper and lower indices equal and using the summation convention. For example, we might wish to define a vector, ""Ai"", from the rank 3 tensor ""Aijk"" by contracting the second and third indices thus, ""Ai = Aijj"" .We also talk of contracting an index of one tensor quantity with an index of another. The contraction of the second index of ""Bij"" with the first of ""Cij"" is a rank 2 tensor, ""BijC jk"".
The //symmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally symmetric// if it is equal to its symmetric part.
The //antisymmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally antisymmetric// if it is equal to its antisymmetric part. A tensor is the sum of its symmetric and antisymmetric parts. It is easily shown that symmetry and antisymmetry are preserved by coordinate transformations — if a tensor is (anti)symmetric in one frame, it is (anti)symmetric in any frame.
The Levi-Civita tensor is defined in ""Minkowski Coordinates"" to be a totally anti-symmetric tensor, with ""ε0123 = 1"". Because of antisymmetry, ""εijkl = 0"" whenever any two indices are the same, and ""εijkl = ±1"" whenever all the indices are different. We have
>>""Definition: A field is a function on a coordinate system or a manifold."">> A tensor valued function on a ""manifold"" is a [[http://en.wikipedia.org/wiki/Tensor_field tensor field]]. A vector field is a rank 1 tensor field. The value of a tensor field at a particular point is a member of a vector or tensor space defined at that point. If we add components of a tensor defined at one point with those defined at another, we are adding members of different tensor spaces. The result is not a tensor. In particular, the ""partial derivative"" of a tensor field, ""x → f(x)"",
for a small displacement ""dxi"" in the ""i""-direction, involves the subtraction of a tensor, ""f(x)"", defined at ""x"", from one, ""f(x + dxi)"", defined at ""x + dxi"". The partial derivative of a tensor field is not, in general, a tensor.
>>""Definition: An equation is covariant if it has the same form in any coordinates, up to the naming of coordinate axes."">>The ""general principle of relativity"" states //Local laws of physics are the same irrespective of the coordinate system used to quantify them.// Vectors are not invariant, as their coordinate representation changes with the coordinate system. Relationships between vectors, defined from the dot product, are unchanged by coordinate transformation. Such relationships are said to be //covariant//. Similarly, relationships between tensors are covariant. The form of the general principle of relativity most directly applicable to classical physics is the principle of general covariance, //The equations of physics have tensorial form//.
To apply the principle of ""general covariance"" in the formulation of physical law we have to find vector and tensor quantities suitable for the description of physics. The 4-vector quantity describing a charged particle is current, ""J"". ""J"" contains information about both-the charge and the motion of the particle. For a stationary charge the 3-current is zero, so the 4-current is ""(q, 0)"". The general form of ""J"" is found by ""Lorentz transformation"" and has the form ""J = (γq, γqv)"", where ""γ"" is the ""Lorentz factor"".
>>""Definition: The electromagnetic field is described by the Faraday tensor, F."">>To describe the electromagnetic force acting on a charged particle we need to contract ""J"" with a tensor representing the electromagnetic field. This tensor is called [[http://en.wikipedia.org/wiki/Electromagnetic_tensor Faraday]], ""F"". The result of contracting ""J"" with ""F"" is a vector, force. So, Faraday is a rank 2 tensor. Faraday should express both the force acting on a charged particle, and the equal and opposite reactive force exerted by the particle on its environment. If Faraday is an antisymmetric tensor, contracting with one index will give the force on the particle and contracting with the other will give the reactive force. We write down the 4-vector law of force:
The simplest situation is a static electric field. In this case the 3-vector force is ""qE = (qEx, qEy, qEz)"". We can then write the Faraday tensor, using antisymmetry to determine the other components.
The Lorentz transformation for a ""boost"" in the ""1""-direction (i.e the ""x""-direction),
Contracting with a 4-current ""( Jt, Jx, Jy, Jz)"" gives a relativistic adjustment to the electrical force acting on the charge ""Jt"", and also introduces a force in the ""x""-direction (the direction opposite to the boost) with terms proportional to ""Jy"" and ""Jz"", i.e. perpendicular to the components of current in those directions. We identify this with magnetic force.
In the general case, Faraday is not represented simply through boosting a static field, but it is the result of fields generated by many particles, each one of which could be regarded as static in the rest frame of that particle. This would lead to a complicated expression, and we write Faraday using the resultant electric and magnetic fields, ""E = (Ex, Ey, Ez)"" and ""B = (Bx, By, Bz)"",
Additions:
<<""Definition: A tensor space is a vector space built from other vector spaces using three simple rules:""
""Rule 1. If xi and y j are vectors, then the object consisting of the products of the coefficients, xiy j, is a tensor.""
""Rule 2. If xij and yij are tensors and a and b are scalars, then axij + byij is a tensor.""
""Rule 3. Addition and multiplication by scalars obey the rules of vector space.""
""""{{image class="right" alt="Tensors-7" title="Basis tensors" url="images/tensors/Tensors-7N.gif"}}For any basis vector, """", in an ""n""-dimensional vector space, ""V1"", and any basis vector, """", in an ""m""-dimensional vector space, ""V2"", make a new object """" (""V1"", and ""V2"" may be the same vector space). The ""mn"" objects of the form """" can be represented as ""mn × 1"" column vectors, each with a ""1"" in one position and a zero in all others (this is how they are represented in computer memory). Introduce the usual rules of matrix addition and multiplication by scalars. The ""mn""-dimensional vector space built like this is a tensor space. Call this tensor space ""T"". The objects in ""T"" are tensors. The objects """" are a basis for ""T"". We write """".
<<""Definition: is the tensor (or outer) product.""
""""
in ""V1"" and
""""
in ""V2"", there is an object in ""T"":
""""
Rank 2 tensors are formed from the product of two vector spaces, and have coefficents with two vector indices. One may find the tensor product of any number of vector spaces in exactly the same way. Rank ""n"" tensors have coefficients with ""n"" indices.
<<""Definition: The tensor product of n vector spaces, V2, … Vn"",
""is a rank n tensor space.""
Tensors are built from vectors, and vector properties apply to each index of a tensor individually. We can apply the index lowering operator, or ""metric"", ""gij"" to change a contravariant index to a covariant one, and the raising operator ""gij"" to restore it to contravariance, e.g. for a Rank 3 tensor, ""Aijk"", we have,
Other indexed quantities appear, known as //non-tensors//. One can raise and lower the indices of non-tensors, using ""g"", but non-tensors are distinguished from tensors because their indices do not satisfy vector transformation laws (any indexed quantity satisfying vector transformation laws can be written as a sum of basis elements and is therefore a tensor). Some treatments even “define” tensors by their transformation properties. In my view, that obscures the mathematical structure of a tensor space.
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor.
//Proof//: since ""xiAijk"" is a tensor
This holds for all ""xi"". So,
So, ""Aijk"" obeys the tensor transformation law. The quotient theorem holds if ""Aijk"" is replaced with a quantity with any number of upstairs or downstairs indices.
To preserve the scalar product under general coordinate transformation we require that, for any vectors ""x"" and ""y"",
Since this is true for all values of ""xi'"" and ""y j'"", we have
Thus, the transformation law for ""gij"" is as for covariant vectors for each suffix. Similarly, the transformation law for ""gij"" is as for contravariant vectors for each superfix. Thus, the ""metric"", ""g"", is a tensor, known as the //metric tensor//.
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\s/span>"", ""g11 = g22 = g22 = −1<\s/span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\s/span>"", ""g11 = −1<\s/span>"", ""g22 = −r^2<\sup><\s/span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\s/span>"" and all other entries equal to zero.
A lower rank tensor is sometimes created from a higher one by //contraction//. This means setting upper and lower indices equal and using the summation convention. For example, we might wish to define a vector, ""Ai"", from the rank 3 tensor ""Aijk"" by contracting the second and third indices thus, ""Ai = Aijj"" .We also talk of contracting an index of one tensor quantity with an index of another. The contraction of the second index of ""Bij"" with the first of ""Cij"" is a rank 2 tensor, ""BijC jk"".
The //symmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally symmetric// if it is equal to its symmetric part.
The //antisymmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally antisymmetric// if it is equal to its antisymmetric part. A tensor is the sum of its symmetric and antisymmetric parts. It is easily shown that symmetry and antisymmetry are preserved by coordinate transformations — if a tensor is (anti)symmetric in one frame, it is (anti)symmetric in any frame.
The Levi-Civita tensor is defined in ""Minkowski Coordinates"" to be a totally anti-symmetric tensor, with ""ε0123 = 1"". Because of antisymmetry, ""εijkl = 0"" whenever any two indices are the same, and ""εijkl = ±1"" whenever all the indices are different. We have
>>""Definition: A field is a function on a coordinate system or a manifold."">> A tensor valued function on a ""manifold"" is a [[http://en.wikipedia.org/wiki/Tensor_field tensor field]]. A vector field is a rank 1 tensor field. The value of a tensor field at a particular point is a member of a vector or tensor space defined at that point. If we add components of a tensor defined at one point with those defined at another, we are adding members of different tensor spaces. The result is not a tensor. In particular, the ""partial derivative"" of a tensor field, ""x → f(x)"",
for a small displacement ""dxi"" in the ""i""-direction, involves the subtraction of a tensor, ""f(x)"", defined at ""x"", from one, ""f(x + dxi)"", defined at ""x + dxi"". The partial derivative of a tensor field is not, in general, a tensor.
>>""Definition: An equation is covariant if it has the same form in any coordinates, up to the naming of coordinate axes."">>The ""general principle of relativity"" states //Local laws of physics are the same irrespective of the coordinate system used to quantify them.// Vectors are not invariant, as their coordinate representation changes with the coordinate system. Relationships between vectors, defined from the dot product, are unchanged by coordinate transformation. Such relationships are said to be //covariant//. Similarly, relationships between tensors are covariant. The form of the general principle of relativity most directly applicable to classical physics is the principle of general covariance, //The equations of physics have tensorial form//.
To apply the principle of ""general covariance"" in the formulation of physical law we have to find vector and tensor quantities suitable for the description of physics. The 4-vector quantity describing a charged particle is current, ""J"". ""J"" contains information about both-the charge and the motion of the particle. For a stationary charge the 3-current is zero, so the 4-current is ""(q, 0)"". The general form of ""J"" is found by ""Lorentz transformation"" and has the form ""J = (γq, γqv)"", where ""γ"" is the ""Lorentz factor"".
>>""Definition: The electromagnetic field is described by the Faraday tensor, F."">>To describe the electromagnetic force acting on a charged particle we need to contract ""J"" with a tensor representing the electromagnetic field. This tensor is called [[http://en.wikipedia.org/wiki/Electromagnetic_tensor Faraday]], ""F"". The result of contracting ""J"" with ""F"" is a vector, force. So, Faraday is a rank 2 tensor. Faraday should express both the force acting on a charged particle, and the equal and opposite reactive force exerted by the particle on its environment. If Faraday is an antisymmetric tensor, contracting with one index will give the force on the particle and contracting with the other will give the reactive force. We write down the 4-vector law of force:
The simplest situation is a static electric field. In this case the 3-vector force is ""qE = (qEx, qEy, qEz)"". We can then write the Faraday tensor, using antisymmetry to determine the other components.
The Lorentz transformation for a ""boost"" in the ""1""-direction (i.e the ""x""-direction),
Contracting with a 4-current ""( Jt, Jx, Jy, Jz)"" gives a relativistic adjustment to the electrical force acting on the charge ""Jt"", and also introduces a force in the ""x""-direction (the direction opposite to the boost) with terms proportional to ""Jy"" and ""Jz"", i.e. perpendicular to the components of current in those directions. We identify this with magnetic force.
In the general case, Faraday is not represented simply through boosting a static field, but it is the result of fields generated by many particles, each one of which could be regarded as static in the rest frame of that particle. This would lead to a complicated expression, and we write Faraday using the resultant electric and magnetic fields, ""E = (Ex, Ey, Ez)"" and ""B = (Bx, By, Bz)"",
Deletions:
<<""Definition: A tensor space is a vector space built from other vector spaces using three simple rules:""
""Rule 1. If xi and y j are vectors, then the object consisting of the products of the coefficients, xiy j, is a tensor.""
""Rule 2. If xij and yij are tensors and a and b are scalars, then axij + byij is a tensor.""
""Rule 3. Addition and multiplication by scalars obey the rules of vector space.""
""""{{image class="right" alt="Tensors-7" title="Basis tensors" url="images/tensors/Tensors-7N.gif"}}For any basis vector, """", in an ""n""-dimensional vector space, ""V1"", and any basis vector, """", in an ""m""-dimensional vector space, ""V2"", make a new object """" (""V1"", and ""V2"" may be the same vector space). The ""mn"" objects of the form """" can be represented as ""mn × 1"" column vectors, each with a ""1"" in one position and a zero in all others (this is how they are represented in computer memory). Introduce the usual rules of matrix addition and multiplication by scalars. The ""mn""-dimensional vector space built like this is a tensor space. Call this tensor space ""T"". The objects in ""T"" are tensors. The objects """" are a basis for ""T"". We write """".
<<""Definition: is the tensor (or outer) product.""
""""
in ""V1"" and
""""
in ""V2"", there is an object in ""T"":
""""
Rank 2 tensors are formed from the product of two vector spaces, and have coefficents with two vector indices. One may find the tensor product of any number of vector spaces in exactly the same way. Rank ""n"" tensors have coefficients with ""n"" indices.
<<""Definition: The tensor product of n vector spaces, V2, … Vn"",
""is a rank n tensor space.""
Tensors are built from vectors, and vector properties apply to each index of a tensor individually. We can apply the index lowering operator, or ""metric"", ""gij"" to change a contravariant index to a covariant one, and the raising operator ""gij"" to restore it to contravariance, e.g. for a Rank 3 tensor, ""Aijk"", we have,
Other indexed quantities appear, known as //non-tensors//. One can raise and lower the indices of non-tensors, using ""g"", but non-tensors are distinguished from tensors because their indices do not satisfy vector transformation laws (any indexed quantity satisfying vector transformation laws can be written as a sum of basis elements and is therefore a tensor). Some treatments even “define” tensors by their transformation properties. In my view, that obscures the mathematical structure of a tensor space.
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor.
//Proof//: since ""xiAijk"" is a tensor
This holds for all ""xi"". So,
So, ""Aijk"" obeys the tensor transformation law. The quotient theorem holds if ""Aijk"" is replaced with a quantity with any number of upstairs or downstairs indices.
To preserve the scalar product under general coordinate transformation we require that, for any vectors ""x"" and ""y"",
Since this is true for all values of ""xi'"" and ""y j'"", we have
Thus, the transformation law for ""gij"" is as for covariant vectors for each suffix. Similarly, the transformation law for ""gij"" is as for contravariant vectors for each superfix. Thus, the ""metric"", ""g"", is a tensor, known as the //metric tensor//.
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
A lower rank tensor is sometimes created from a higher one by //contraction//. This means setting upper and lower indices equal and using the summation convention. For example, we might wish to define a vector, ""Ai"", from the rank 3 tensor ""Aijk"" by contracting the second and third indices thus, ""Ai = Aijj"" .We also talk of contracting an index of one tensor quantity with an index of another. The contraction of the second index of ""Bij"" with the first of ""Cij"" is a rank 2 tensor, ""BijC jk"".
The //symmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally symmetric// if it is equal to its symmetric part.
The //antisymmetric part// is found by dividing by the number of terms, i.e. ""n!"" if there are ""n"" indices. A tensor is //totally antisymmetric// if it is equal to its antisymmetric part. A tensor is the sum of its symmetric and antisymmetric parts. It is easily shown that symmetry and antisymmetry are preserved by coordinate transformations — if a tensor is (anti)symmetric in one frame, it is (anti)symmetric in any frame.
The Levi-Civita tensor is defined in ""Minkowski Coordinates"" to be a totally anti-symmetric tensor, with ""ε0123 = 1"". Because of antisymmetry, ""εijkl = 0"" whenever any two indices are the same, and ""εijkl = ±1"" whenever all the indices are different. We have
>>""Definition: A field is a function on a coordinate system or a manifold."">> A tensor valued function on a ""manifold"" is a [[http://en.wikipedia.org/wiki/Tensor_field tensor field]]. A vector field is a rank 1 tensor field. The value of a tensor field at a particular point is a member of a vector or tensor space defined at that point. If we add components of a tensor defined at one point with those defined at another, we are adding members of different tensor spaces. The result is not a tensor. In particular, the ""partial derivative"" of a tensor field, ""x → f(x)"",
for a small displacement ""dxi"" in the ""i""-direction, involves the subtraction of a tensor, ""f(x)"", defined at ""x"", from one, ""f(x + dxi)"", defined at ""x + dxi"". The partial derivative of a tensor field is not, in general, a tensor.
>>""Definition: An equation is covariant if it has the same form in any coordinates, up to the naming of coordinate axes."">>The ""general principle of relativity"" states //Local laws of physics are the same irrespective of the coordinate system used to quantify them.// Vectors are not invariant, as their coordinate representation changes with the coordinate system. Relationships between vectors, defined from the dot product, are unchanged by coordinate transformation. Such relationships are said to be //covariant//. Similarly, relationships between tensors are covariant. The form of the general principle of relativity most directly applicable to classical physics is the principle of general covariance, //The equations of physics have tensorial form//.
To apply the principle of ""general covariance"" in the formulation of physical law we have to find vector and tensor quantities suitable for the description of physics. The 4-vector quantity describing a charged particle is current, ""J"". ""J"" contains information about both-the charge and the motion of the particle. For a stationary charge the 3-current is zero, so the 4-current is ""(q, 0)"". The general form of ""J"" is found by ""Lorentz transformation"" and has the form ""J = (γq, γqv)"", where ""γ"" is the ""Lorentz factor"".
>>""Definition: The electromagnetic field is described by the Faraday tensor, F."">>To describe the electromagnetic force acting on a charged particle we need to contract ""J"" with a tensor representing the electromagnetic field. This tensor is called [[http://en.wikipedia.org/wiki/Electromagnetic_tensor Faraday]], ""F"". The result of contracting ""J"" with ""F"" is a vector, force. So, Faraday is a rank 2 tensor. Faraday should express both the force acting on a charged particle, and the equal and opposite reactive force exerted by the particle on its environment. If Faraday is an antisymmetric tensor, contracting with one index will give the force on the particle and contracting with the other will give the reactive force. We write down the 4-vector law of force:
The simplest situation is a static electric field. In this case the 3-vector force is ""qE = (qEx, qEy, qEz)"". We can then write the Faraday tensor, using antisymmetry to determine the other components.
The Lorentz transformation for a ""boost"" in the ""1""-direction (i.e the ""x""-direction),
Contracting with a 4-current ""( Jt, Jx, Jy, Jz)"" gives a relativistic adjustment to the electrical force acting on the charge ""Jt"", and also introduces a force in the ""x""-direction (the direction opposite to the boost) with terms proportional to ""Jy"" and ""Jz"", i.e. perpendicular to the components of current in those directions. We identify this with magnetic force.
In the general case, Faraday is not represented simply through boosting a static field, but it is the result of fields generated by many particles, each one of which could be regarded as static in the rest frame of that particle. This would lead to a complicated expression, and we write Faraday using the resultant electric and magnetic fields, ""E = (Ex, Ey, Ez)"" and ""B = (Bx, By, Bz)"",
Additions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
Deletions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", """"the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
Additions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", """"the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
Deletions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
Additions:
It is always possible to choose an ""orthogonal basis"". In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime"" has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
Deletions:
It is always possible to choose an ""orthogonal basis""orthornormal basis. In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
Additions:
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor.
So, ""Aijk"" obeys the tensor transformation law. The quotient theorem holds if ""Aijk"" is replaced with a quantity with any number of upstairs or downstairs indices.
Thus, the transformation law for ""gij"" is as for covariant vectors for each suffix. Similarly, the transformation law for ""gij"" is as for contravariant vectors for each superfix. Thus, the ""metric"", ""g"", is a tensor, known as the //metric tensor//.
It is always possible to choose an ""orthogonal basis""orthornormal basis. In an orthogonal basis the metric is a represented by a diagonal matrix. For example, In ""Minkowski Coordinates"", ""the metric for Minkowski spacetime has ""g00 = 1<\span>"", ""g11 = g22 = g22 = −1<\span>"", and all other entries equal to zero. In ""spherical coordinates "", Minkowski spacetime has metric with ""g00 = 1<\span>"", ""g11 = −1<\span>"", ""g22 = −r^2<\sup><\span>"", ""g22 = −r^2<\sup>sin2<\sup>;theta;<\span>"" and all other entries equal to zero.
The Levi-Civita tensor is defined in ""Minkowski Coordinates"" to be a totally anti-symmetric tensor, with ""ε0123 = 1"". Because of antisymmetry, ""εijkl = 0"" whenever any two indices are the same, and ""εijkl = ±1"" whenever all the indices are different. We have
Deletions:
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor.
So, ""Aijk"" obeys the tensor transformation law. This holds if ""Aijk"" is replaced with a quantity with any number of upstairs or downstairs indices.
Thus, the transformation law for ""gij"" is as for covariant vectors for each suffix. Similarly, the transformation law for ""gij"" is as for contravariant vectors for each superfix. Thus, the metric, ""g"", is a tensor, known as the //metric tensor//.
The Levi-Civita tensor is defined in Minkowski coordinates to be a totally anti-symmetric tensor, with ""ε0123 = 1"". Because of antisymmetry, ""εijkl = 0"" whenever any two indices are the same, and ""εijkl = ±1"" whenever all the indices are different. We have
Additions:
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor.
//Proof//: since ""xiAijk"" is a tensor
Deletions:
The //quotient theorem// states that if, for any vector ""xi"", ""xiAijk"" is a tensor, then ""Aijk"" is a tensor. //Proof//: since ""xiAijk"" is a tensor
Additions:
