Lorentz scalar

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In physics, a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar may be generated from multiplication of vectors or tensors. While the components of vectors and tensors are in general altered by Lorentz transformations, scalars remain unchanged.

A Lorentz scalar is not necessarily a scalar in the strict sense of being a (0,0)-tensor, that is, invariant under any base transformation. For example, the determinant of the matrix of base vectors is a number that is invariant under Lorentz transformations, but it is not invariant under any base transformation.

Simple scalars in special relativity[edit]

The length of a position vector[edit]

World lines for two particles at different speeds.

In special relativity the location of a particle in 4-dimensional spacetime is given by

 x^{\mu} = (ct, \mathbf{x} )

where \mathbf{x} = \mathbf{v} t is the position in 3-dimensional space of the particle, \mathbf{v} is the velocity in 3-dimensional space and  c is the speed of light.

The "length" of the vector is a Lorentz scalar and is given by

 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 \tau is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by

\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 is a time-like metric.

Often the alternate signature of the Minkowski metric is used in which the signs of the ones are reversed.

\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 is a space-like metric.

In the Minkowski metric the space-like interval  s is defined as

 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 the space-like Minkowski metric in the rest of this article.

The length of a velocity vector[edit]

The velocity vectors in spacetime for a particle at two different speeds. In relativity an acceleration is equivalent to a rotation in spacetime

The velocity in spacetime is defined as

 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

 \gamma \ \stackrel{\mathrm{def}}{=}\   { 1 \over {\sqrt {1 - {{\mathbf{v} \cdot \mathbf{v} } \over c^2} } } } .

The magnitude of the 4-velocity is a Lorentz scalar,

 v_{\mu} v^{\mu} = -c^2\,.

Hence, c is a Lorentz scalar.

The inner product of acceleration and velocity[edit]

The 4-acceleration is given by

 a^{\mu} \ \stackrel{\mathrm{def}}{=}\  {dv^{\mu} \over d\tau} .

The 4-acceleration is always perpendicular to the 4-velocity

 0 = {1 \over 2} {d \over d\tau} \left ( v_{\mu}v^{\mu} \right ) = {d v_{\mu} \over d\tau}  v^{\mu} = a_{\mu} v^{\mu} .

Therefore, we can regard acceleration in spacetime as simply 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 \over d\tau} = \mathbf{F} \cdot { \mathbf{v}}

where  E is the energy of a particle and  \mathbf{F} is the 3-force on the particle.

Energy, rest mass, 3-momentum, and 3-speed from 4-momentum[edit]

The 4-momentum of a particle is

 p^{\mu} = m v^{\mu} =   \left ( \gamma m c,  \gamma {  m \mathbf{v} }  \right ) = \left ( \gamma m c,   {   \mathbf{p} }  \right ) = \left ( {E \over c } ,   {   \mathbf{p} }  \right )

where  m is the particle rest mass,  \mathbf{p} is the momentum in 3-space, and

 E = \gamma m c^2 \,

is the energy of the particle.

Measurement of the energy of a particle[edit]

Consider a second particle with 4-velocity  u and a 3-velocity   \mathbf{u}_2 . In the rest frame of the second particle the inner product of  u with  p is proportional to the energy of the first particle

 p_{\mu} u^{\mu} = -  { E_1}

where the subscript 1 indicates the first particle.

Since the relationship is true in the rest frame of the second particle, it is true in any reference frame.  E_1 , the energy of the first particle in the frame of the second particle, is a Lorentz scalar. Therefore

  { 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 is still the energy of the first particle in the frame of the second particle .

Measurement of the rest mass of the particle[edit]

In the rest frame of the particle the inner product of the momentum is

 p_{\mu} p^{\mu} = - (mc)^2  \,.

Therefore the rest mass (m) is a Lorentz scalar. The relationship remains true independent of the frame in which the inner product is calculated. In many cases the rest mass is written as  m_{0} to avoid confusion with the relativistic mass, which is  \gamma m_{0}

Measurement of the 3-momentum of the particle[edit]

Note that

  \left ( 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 the magnitude of the 3-momentum of the particle as measured in the frame of the second particle is a Lorentz scalar.

Measurement of the 3-speed of the particle[edit]

The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars

 v_1^2 = \mathbf{v}_1 \cdot \mathbf{v}_1  = { { \mathbf{p}_1 \cdot \mathbf{p}_1 c^4 } \over  { E_1^2  } } .

More complicated scalars[edit]

Scalars may also be constructed from the tensors and vectors, from the contraction of tensors, or combinations of contractions of tensors and vectors.

References[edit]

  • Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0. 
  • Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.