Presented in this paper the new relativistic formulas for kinetic, rest and total energies differ from the commonly applied analogous expressions. The new relativistic kinetic energy and its relation with the spatial part of the four-momentum are given by the expressions similar to its classical equivalents. In particular, I justified that the rest energy is half of the amount determined by the traditional relation. The most popular formula of physics (mass-energy equivalence principle) is not correct.
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Zbigniew Osiak
ENERGY
IN
SPECIAL RELATIVITY
Zbigniew Osiak
EҭERGY
Iҭ
SPECIAL RELATIVITY
Energy in Special Relativity
By Zbigniew Osiak
Self Publishing
© Copyright 2011, 214 by Zbigniew Osiak
All rights reserved
ISBN: 978-83-272-3448-3
Portrait of the author by Rafał Pudło
e-mail: zbigniew.osiak@gmail.com
Energy in Special Relativity
Zbigniew Osiak
The Wroclaw College of Humanities, Wroclaw, Poland
The University of the Third Age in the Wroclaw University, Wroclaw,
Poland
E-mail: zbigniew.osiak@gmail.com
Abstract
I give new relativistic formulas for kinetic, rest and total energies.
The change in kinetic energy of a particle will be determined as the
work done by the spatial part of the Minkowski four-force. I present
a new relation between the relativistic kinetic energy and the spa-
tial part of the four-momentum also interpretation of the temporal
component of the Minkowski four-force.
Contents
Abstract
1 Introduction
2 The Lorentz covariant four-dimensional
Minkowski equations of motion
3 The spatial part of the Minkowski four-force
4 The new formulas for energies in relativistic mechanics
5 The square modulus of the four-velocity
6 The new relation between kinetic energy and momentum
7 The temporal component of the Minkowski four-force
8 A more general form of the Mikowski equations of motion
1
1
2
2
3
3
4
4
5
6
Zbigniew Osiak
2
Energy in Special Relativity
9 Conclusions and discussion
References
1 Introduction
6
7
The traditional form of the mass-energy equivalence principle proposed by
Einstein [1] E0 = mc2 is sometimes derived [2] using the Planck equations of
motion [3] F = d(mγv)/dt. Usage of these equations in special relativity leads
to errors because they are not covariant under Lorentz transformations. The
Lorentz covariance of the physical equations and the Lorentz invariance of the
speed of light in a vacuum are the two basic postulates of special relativity.
In my view, the Planck equations of motion are an interesting example of
a heuristic hypothesis and have only historical significance. For calculation
of the relativistic kinetic energy and its relationship with the rest and the
total energies it is necessary to use the Lorentz covariant four-dimensional
Minkowski equations of motion [4].
2 The Lorentz covariant four-dimensional
Minkowski equations of motion
The Lorentz covariant four-dimensional Minkowski equations of motion are
given by:
˜Fα = mγ
d˜vα
dt
= m˜aα
(1)
where
˜Fα – components of the Minkowski four-force; α = 1, 2, 3, 4; m – invariant
mass of a particle; γ ≡ (1 − v2c−2)
2 – Lorentz factor; c – speed of light
in vacuum; ˜vα ≡ γ (dxα/dt) – components of the four-velocity; x1 ≡ x,
x2 ≡ y, x3 ≡ z, x4 ≡ ict; i – imaginary unit; v ≡ dr/dt – three-dimensional
velocity; r ≡ (x, y, z) = (x1, x2, x3) – three-dimensional position vector; v2 ≡
(dx1/dt)2 + (dx2/dt)2 + (dx3/dt)2 – square modulus of the three-dimensional
velocity; ˜aα ≡ γ (d˜vα/dt) – components of the four-acceleration.
− 1
Self Publishing
(28.12.2011)
www.virtualo.pl
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