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47-402: The principle of mutability is the notion that any physical property which appears to follow a conservation law may undergo some physical process that violates its conservation. John Archibald Wheeler offered this speculative principle after Stephen Hawking predicted the evaporation of black holes which violates baryon number conservation . This article about theoretical physics

94-402: A ( y ) y x {\displaystyle j_{x}=j_{y}(y)y_{x}=a(y)y_{x}} the conservation equation can be put into the current density form: y t + j x ( y ) = 0 {\displaystyle y_{t}+j_{x}(y)=0} In a space with more than one dimension the former definition can be extended to an equation that can be put into

141-401: A rotation of the 4-velocity. The inner product of the acceleration and the velocity is a Lorentz scalar and is zero. This rotation is simply an expression of energy conservation: d E d τ = F ⋅ v {\displaystyle {dE \over d\tau }=\mathbf {F} \cdot \mathbf {v} } where E {\displaystyle E} is

188-543: Is CPT symmetry , the simultaneous inversion of space and time coordinates, together with swapping all particles with their antiparticles; however being a discrete symmetry Noether's theorem does not apply to it. Accordingly, the conserved quantity, CPT parity , can usually not be meaningfully calculated or determined. There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions. The total amount of some conserved quantity in

235-402: Is a stub . You can help Misplaced Pages by expanding it . Conservation law A local conservation law is usually expressed mathematically as a continuity equation , a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by

282-693: Is a space-like metric. In the Minkowski metric the space-like interval s {\displaystyle s} is defined as x μ x μ = η μ ν x μ x ν = x ⋅ x − ( c t ) 2   = d e f   s 2 . {\displaystyle x_{\mu }x^{\mu }=\eta _{\mu \nu }x^{\mu }x^{\nu }=\mathbf {x} \cdot \mathbf {x} -(ct)^{2}\ {\stackrel {\mathrm {def} }{=}}\ s^{2}.} We use

329-685: Is a time-like metric. Often the alternate signature of the Minkowski metric is used in which the signs of the ones are reversed. η μ ν = η μ ν = ( − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) . {\displaystyle \eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}.} This

376-543: Is always perpendicular to the 4-velocity 0 = 1 2 d d τ ( v μ v μ ) = d v μ d τ v μ = a μ v μ . {\displaystyle 0={1 \over 2}{d \over d\tau }\left(v_{\mu }v^{\mu }\right)={dv_{\mu } \over d\tau }v^{\mu }=a_{\mu }v^{\mu }.} Therefore, we can regard acceleration in spacetime as simply

423-615: Is called the conserved ( vector ) quantity, ∇ y is its gradient , 0 is the zero vector , and A ( y ) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A ( y ) usually corresponding to the Jacobian of a current density matrix J ( y ) : A ( y ) = J y ( y ) {\displaystyle \mathbf {A} (\mathbf {y} )=\mathbf {J} _{\mathbf {y} }(\mathbf {y} )} and

470-401: Is not Lorentz invariant , so phenomena like the above do not occur in nature. Due to special relativity , if the appearance of the energy at A and disappearance of the energy at B are simultaneous in one inertial reference frame , they will not be simultaneous in other inertial reference frames moving with respect to the first. In a moving frame one will occur before the other; either

517-419: Is not a conservation equation but the general kind of balance equation describing a dissipative system . The dependent variable y is called a nonconserved quantity , and the inhomogeneous term s ( y , x , t ) is the- source , or dissipation . For example, balance equations of this kind are the momentum and energy Navier-Stokes equations , or the entropy balance for a general isolated system . In

SECTION 10

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564-458: Is ordinary and the other is an antiparticle. With respect to symmetries and invariance principles, three special conservation laws have been described, associated with inversion or reversal of space, time, and charge. Conservation laws are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and engineering. Most conservation laws are exact, or absolute, in

611-428: Is still the energy of the first particle in the frame of the second particle. In the rest frame of the particle the inner product of the momentum is p μ p μ = − ( m c ) 2 . {\displaystyle p_{\mu }p^{\mu }=-(mc)^{2}\,.} Therefore, the rest mass ( m ) is a Lorentz scalar. The relationship remains true independent of

658-687: Is the density of q (amount per unit volume), j is the flux of q (amount crossing a unit area in unit time), and t is time. If we assume that the motion u of the charge is a continuous function of position and time, then j = ρ u ∂ ρ ∂ t = − ∇ ⋅ ( ρ u ) . {\displaystyle {\begin{aligned}\mathbf {j} &=\rho \mathbf {u} \\{\frac {\partial \rho }{\partial t}}&=-\nabla \cdot (\rho \mathbf {u} )\,.\end{aligned}}} In one space dimension this can be put into

705-441: Is the particle rest mass, p {\displaystyle \mathbf {p} } is the momentum in 3-space, and E = γ m c 2 {\displaystyle E=\gamma mc^{2}} is the energy of the particle. Consider a second particle with 4-velocity u {\displaystyle u} and a 3-velocity u 2 {\displaystyle \mathbf {u} _{2}} . In

752-869: Is the position in 3-dimensional space of the particle, v {\displaystyle \mathbf {v} } is the velocity in 3-dimensional space and c {\displaystyle c} is the speed of light . The "length" of the vector is a Lorentz scalar and is given by x μ x μ = η μ ν x μ x ν = ( c t ) 2 − x ⋅ x   = d e f   ( c τ ) 2 {\displaystyle x_{\mu }x^{\mu }=\eta _{\mu \nu }x^{\mu }x^{\nu }=(ct)^{2}-\mathbf {x} \cdot \mathbf {x} \ {\stackrel {\mathrm {def} }{=}}\ (c\tau )^{2}} where τ {\displaystyle \tau }

799-638: Is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by η μ ν = η μ ν = ( 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1 ) . {\displaystyle \eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}.} This

846-408: The current coefficient , usually corresponding to the partial derivative in the conserved quantity of a current density of the conserved quantity j ( y ) : a ( y ) = j y ( y ) {\displaystyle a(y)=j_{y}(y)} In this case since the chain rule applies: j x = j y ( y ) y x =

893-490: The divergence of a vector current density associated to the conserved quantity j ( y ) : y t + ∇ ⋅ j ( y ) = 0 {\displaystyle y_{t}+\nabla \cdot \mathbf {j} (y)=0} This is the case for the continuity equation : ρ t + ∇ ⋅ ( ρ u ) = 0 {\displaystyle \rho _{t}+\nabla \cdot (\rho \mathbf {u} )=0} Here

940-403: The one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form: y t + a ( y ) y x = 0 {\displaystyle y_{t}+a(y)y_{x}=0} where the dependent variable y ( x , t ) is called the density of the conserved (scalar) quantity, and a ( y ) is called

987-538: The outer product . Conservation equations can usually also be expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form , extending the class of admissible solutions to include discontinuous solutions. By integrating in any space-time domain the current density form in 1-D space: y t + j x ( y ) = 0 {\displaystyle y_{t}+j_{x}(y)=0} and by using Green's theorem ,

SECTION 20

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1034-525: The uniformity of time and the conservation of angular momentum arises from the isotropy of space , i.e. because there is no preferred direction of space. Notably, there is no conservation law associated with time-reversal , although more complex conservation laws combining time-reversal with other symmetries are known. A partial listing of physical conservation equations due to symmetry that are said to be exact laws , or more precisely have never been proven to be violated: Another exact symmetry

1081-608: The 4-velocity is a Lorentz scalar, v μ v μ = − c 2 . {\displaystyle v_{\mu }v^{\mu }=-c^{2}\,.} Hence, ⁠ c {\displaystyle c} ⁠ is a Lorentz scalar. The 4-acceleration is given by a μ   = d e f   d v μ d τ . {\displaystyle a^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {dv^{\mu } \over d\tau }.} The 4-acceleration

1128-476: The Universe remains constant. All of the conservation laws listed above are local conservation laws. A local conservation law is expressed mathematically by a continuity equation , which states that the change in the quantity in a volume is equal to the total net "flux" of the quantity through the surface of the volume. The following sections discuss continuity equations in general. In continuum mechanics ,

1175-407: The amount of the quantity which flows in or out of the volume. From Noether's theorem , every differentiable symmetry leads to a conservation law. Other conserved quantities can exist as well. Conservation laws are fundamental to our understanding of the physical world, in that they describe which processes can or cannot occur in nature. For example, the conservation law of energy states that

1222-778: The conservation equation can be put into the form: y t + ∇ ⋅ J ( y ) = 0 {\displaystyle \mathbf {y} _{t}+\nabla \cdot \mathbf {J} (\mathbf {y} )=\mathbf {0} } For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are: ∇ ⋅ u = 0 , ∂ u ∂ t + u ⋅ ∇ u + ∇ s = 0 , {\displaystyle \nabla \cdot \mathbf {u} =0\,,\qquad {\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} +\nabla s=\mathbf {0} ,} where: It can be shown that

1269-599: The conserved (vector) quantity and the current density matrix for these equations are respectively: y = ( 1 u ) ; J = ( u u ⊗ u + s I ) ; {\displaystyle {\mathbf {y} }={\begin{pmatrix}1\\\mathbf {u} \end{pmatrix}};\qquad {\mathbf {J} }={\begin{pmatrix}\mathbf {u} \\\mathbf {u} \otimes \mathbf {u} +s\mathbf {I} \end{pmatrix}};\qquad } where ⊗ {\displaystyle \otimes } denotes

1316-546: The conserved quantity is the mass , with density ρ ( r , t ) and current density ρ u , identical to the momentum density , while u ( r , t ) is the flow velocity . In the general case a conservation equation can be also a system of this kind of equations (a vector equation ) in the form: y t + A ( y ) ⋅ ∇ y = 0 {\displaystyle \mathbf {y} _{t}+\mathbf {A} (\mathbf {y} )\cdot \nabla \mathbf {y} =\mathbf {0} } where y

1363-683: The corresponding Lorentz transformation. Other examples of Lorentz scalars are the "length" of 4-velocities (see below), or the Ricci curvature in a point in spacetime from general relativity , which is a contraction of the Riemann curvature tensor there. In special relativity the location of a particle in 4-dimensional spacetime is given by x μ = ( c t , x ) {\displaystyle x^{\mu }=(ct,\mathbf {x} )} where x = v t {\displaystyle \mathbf {x} =\mathbf {v} t}

1410-460: The energy at A will appear before or after the energy at B disappears. In both cases, during the interval energy will not be conserved. A stronger form of conservation law requires that, for the amount of a conserved quantity at a point to change, there must be a flow, or flux of the quantity into or out of the point. For example, the amount of electric charge at a point is never found to change without an electric current into or out of

1457-632: The energy of a particle and F {\displaystyle \mathbf {F} } is the 3-force on the particle. The 4-momentum of a particle is p μ = m v μ = ( γ m c , γ m v ) = ( γ m c , p ) = ( E c , p ) {\displaystyle p^{\mu }=mv^{\mu }=\left(\gamma mc,\gamma m\mathbf {v} \right)=\left(\gamma mc,\mathbf {p} \right)=\left({\frac {E}{c}},\mathbf {p} \right)} where m {\displaystyle m}

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1504-568: The form of a homogeneous first-order quasilinear hyperbolic equation : y t + A ( y ) y x = 0 {\displaystyle y_{t}+A(y)y_{x}=0} where the dependent variable y is called the density of a conserved quantity , and A ( y ) is called the current Jacobian , and the subscript notation for partial derivatives has been employed. The more general inhomogeneous case: y t + A ( y ) y x = s {\displaystyle y_{t}+A(y)y_{x}=s}

1551-399: The form: y t + a ( y ) ⋅ ∇ y = 0 {\displaystyle y_{t}+\mathbf {a} (y)\cdot \nabla y=0} where the conserved quantity is y ( r , t ) , ⋅ denotes the scalar product , ∇ is the nabla operator, here indicating a gradient , and a ( y ) is a vector of current coefficients, analogously corresponding to

1598-1151: The frame in which the inner product is calculated. In many cases the rest mass is written as m 0 {\displaystyle m_{0}} to avoid confusion with the relativistic mass, which is γ m 0 {\displaystyle \gamma m_{0}} . Note that ( p μ u μ c ) 2 + p μ p μ = E 1 2 c 2 − ( m c ) 2 = ( γ 1 2 − 1 ) ( m c ) 2 = γ 1 2 v 1 ⋅ v 1 m 2 = p 1 ⋅ p 1 . {\displaystyle \left({\frac {p_{\mu }u^{\mu }}{c}}\right)^{2}+p_{\mu }p^{\mu }={E_{1}^{2} \over c^{2}}-(mc)^{2}=\left(\gamma _{1}^{2}-1\right)(mc)^{2}=\gamma _{1}^{2}{\mathbf {v} _{1}\cdot \mathbf {v} _{1}}m^{2}=\mathbf {p} _{1}\cdot \mathbf {p} _{1}.} The square of

1645-569: The integral form is: ∫ − ∞ ∞ y d x + ∫ 0 ∞ j ( y ) d t = 0 {\displaystyle \int _{-\infty }^{\infty }y\,dx+\int _{0}^{\infty }j(y)\,dt=0} In a similar fashion, for the scalar multidimensional space, the integral form is: ∮ [ y d N r + j ( y ) d t ] = 0 {\displaystyle \oint \left[y\,d^{N}r+j(y)\,dt\right]=0} where

1692-858: The line integration is performed along the boundary of the domain, in an anticlockwise manner. Moreover, by defining a test function φ ( r , t ) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition . In 1-D space it is: ∫ 0 ∞ ∫ − ∞ ∞ ϕ t y + ϕ x j ( y ) d x d t = − ∫ − ∞ ∞ ϕ ( x , 0 ) y ( x , 0 ) d x {\displaystyle \int _{0}^{\infty }\int _{-\infty }^{\infty }\phi _{t}y+\phi _{x}j(y)\,dx\,dt=-\int _{-\infty }^{\infty }\phi (x,0)y(x,0)\,dx} In

1739-617: The magnitude of the 3-momentum of the particle as measured in the frame of the second particle is a Lorentz scalar. The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars v 1 2 = v 1 ⋅ v 1 = p 1 ⋅ p 1 E 1 2 c 4 . {\displaystyle v_{1}^{2}=\mathbf {v} _{1}\cdot \mathbf {v} _{1}={\frac {\mathbf {p} _{1}\cdot \mathbf {p} _{1}}{E_{1}^{2}}}c^{4}.} Scalars may also be constructed from

1786-405: The most general form of an exact conservation law is given by a continuity equation . For example, conservation of electric charge q is ∂ ρ ∂ t = − ∇ ⋅ j {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \,} where ∇⋅ is the divergence operator, ρ

1833-407: The point that carries the difference in charge. Since it only involves continuous local changes, this stronger type of conservation law is Lorentz invariant ; a quantity conserved in one reference frame is conserved in all moving reference frames. This is called a local conservation law. Local conservation also implies global conservation; that the total amount of the conserved quantity in

1880-695: The relationship is true in the rest frame of the second particle, it is true in any reference frame. E 1 {\displaystyle E_{1}} , the energy of the first particle in the frame of the second particle, is a Lorentz scalar. Therefore, E 1 = γ 1 γ 2 m 1 c 2 − γ 2 p 1 ⋅ u 2 {\displaystyle E_{1}=\gamma _{1}\gamma _{2}m_{1}c^{2}-\gamma _{2}\mathbf {p} _{1}\cdot \mathbf {u} _{2}} in any inertial reference frame, where E 1 {\displaystyle E_{1}}

1927-417: The rest frame of the second particle the inner product of u {\displaystyle u} with p {\displaystyle p} is proportional to the energy of the first particle p μ u μ = − E 1 {\displaystyle p_{\mu }u^{\mu }=-E_{1}} where the subscript 1 indicates the first particle. Since

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1974-522: The scalar product of vectors, or by contracting tensors. While the components of the contracted quantities may change under Lorentz transformations, the Lorentz scalars remain unchanged. A simple Lorentz scalar in Minkowski spacetime is the spacetime distance ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under

2021-474: The sense that they apply to all possible processes. Some conservation laws are partial, in that they hold for some processes but not for others. One particularly important result concerning conservation laws is Noether's theorem , which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry of the Universe . For example, the conservation of energy follows from

2068-1167: The space-like Minkowski metric in the rest of this article. The velocity in spacetime is defined as v μ   = d e f   d x μ d τ = ( c d t d τ , d t d τ d x d t ) = ( γ c , γ v ) = γ ( c , v ) {\displaystyle v^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {dx^{\mu } \over d\tau }=\left(c{dt \over d\tau },{dt \over d\tau }{d\mathbf {x} \over dt}\right)=\left(\gamma c,\gamma {\mathbf {v} }\right)=\gamma \left(c,{\mathbf {v} }\right)} where γ   = d e f   1 1 − v ⋅ v c 2 . {\displaystyle \gamma \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {1-{{\mathbf {v} \cdot \mathbf {v} } \over c^{2}}}}}.} The magnitude of

2115-471: The total quantity of energy in an isolated system does not change, though it may change form. In general, the total quantity of the property governed by that law remains unchanged during physical processes. With respect to classical physics, conservation laws include conservation of energy, mass (or matter), linear momentum, angular momentum, and electric charge. With respect to particle physics, particles cannot be created or destroyed except in pairs, where one

2162-426: The universe could remain unchanged if an equal amount were to appear at one point A and simultaneously disappear from another separate point B . For example, an amount of energy could appear on Earth without changing the total amount in the Universe if the same amount of energy were to disappear from some other region of the Universe. This weak form of "global" conservation is really not a conservation law because it

2209-427: The weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives. Lorentz invariant In a relativistic theory of physics , a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation . A Lorentz scalar may be generated from, e.g.,

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