Relativistic Mechanics - Force

Force

In special relativity, Newton's second law does not hold in its form F = ma, but it does if it is expressed as

where p = γm0v is the momentum as defined above and m0 is the invariant mass. Thus, the force is given by

Derivation

Starting from

Carrying out the derivatives gives

using the identity

,

gives

If the acceleration is separated into the part parallel to the velocity and the part perpendicular to it, one gets

\begin{align}
\mathbf{F} & = \frac{\gamma^3 m_0 v^{2}}{c^2} \, \mathbf{a}_{\parallel} + \gamma m_0 \, (\mathbf{a}_{\parallel} + \mathbf{a}_{\perp})\\
& = \gamma^3 m_0 \left( \frac{v^2}{c^2} + \frac{1}{\gamma^2} \right) \mathbf{a}_{\parallel} + \gamma m_0 \, \mathbf{a}_{\perp} \\
& = \gamma^3 m_0 \left( \frac{v^{2}}{c^2} + 1 - \frac{v^{2}}{c^2} \right) \mathbf{a}_{\parallel} + \gamma m_0 \, \mathbf{a}_{\perp} \\
& = \gamma^3 m_0 \, \mathbf{a}_{\parallel} + \gamma m_0 \, \mathbf{a}_{\perp}
\end{align}\,

Consequently in some old texts, γ3m0 is referred to as the longitudinal mass, and γm0 is referred to as the transverse mass, which is numerically the same as the relativistic mass. See mass in special relativity.

If one inverts this to calculate acceleration from force, one gets

The force described in this section is the classical 3-D force which is not a four-vector. This 3-D force is the appropriate concept of force since it is the force which obeys Newton's third law of motion. It should not be confused with the so-called four-force which is merely the 3-D force in the comoving frame of the object transformed as if it were a four-vector. However, the density of 3-D force (linear momentum transferred per unit four-volume) is a four-vector (density of weight +1) when combined with the negative of the density of power transferred.

Read more about this topic:  Relativistic Mechanics

Famous quotes containing the word force:

    There is ... but one response possible from us: Force, Force to the uttermost, Force without stint or limit, the righteous and triumphant Force which shall make Right the law of the world and cast every selfish dominion down in the dust.
    Woodrow Wilson (1856–1924)

    The example of America must be the example, not merely of peace because it will not fight, but of peace because it is the healing and elevating influence of the world, and strife is not. There is such a thing as a man being too proud to fight. There is such a thing as a nation being so right that it does not need to convince others by force that it is right.
    Woodrow Wilson (1856–1924)

    A grocer is attracted to his business by a magnetic force as great as the repulsion which renders it odious to artists.
    Honoré De Balzac (1799–1850)