- 306

- 2

I was summarizing for myself the various four-vectors of mechanics:

[tex]

\begin{align*}

x &= ct + \mathbf{x} \\

V &= \frac{dx}{d\tau} = \gamma(c + \mathbf{v}) \\

P &= m V = E/c + \gamma\mathbf{p} \\

f &= m\frac{d^2 x}{d\tau^2} = m\frac{d V}{d\tau} \\

\end{align*}

[/tex]

where:

[tex]

\begin{align*}

\gamma^{-2} &= (1 - {\lvert \mathbf{v}/c \rvert}^2) \\

d\tau &= {\left(\frac{dx}{d\lambda} \cdot \frac{dx}{d\lambda}\right)}^{1/2} d\lambda \\

x \cdot x = {\lvert x \rvert}^2 &= c^2t^2 - {\lvert \mathbf{x} \rvert}^2 \\

E &= \int f \cdot (c d\tau) \\

\mathbf{v} &= \frac{d\mathbf{x}}{dt} \\

\mathbf{p} &= m\mathbf{v} \\

\end{align*}

[/tex]

Invarients for the first three four vectors are:

[tex]

\begin{align*}

{\lvert x \rvert}^2 &= c^2 t^2 - {\lvert \mathbf{x} \rvert}^2 = c^2 \tau^2 \\

{\lvert V \rvert}^2 &= \gamma^2 (c^2 - {\lvert \mathbf{v} \rvert}^2) = c^2 \\

{\lvert P \rvert}^2 &= m^2 {\lvert V \rvert}^2 = m^2 c^2 \\

\end{align*}

[/tex]

Is the minkowski norm of the four vector force:

[tex]

f = m\frac{d^2 x}{d\tau^2}

[/tex]

also an invarient? I think it has to be. Assuming that is the case, what would the value (and significance if any) of this be?

[tex]

\begin{align*}

x &= ct + \mathbf{x} \\

V &= \frac{dx}{d\tau} = \gamma(c + \mathbf{v}) \\

P &= m V = E/c + \gamma\mathbf{p} \\

f &= m\frac{d^2 x}{d\tau^2} = m\frac{d V}{d\tau} \\

\end{align*}

[/tex]

where:

[tex]

\begin{align*}

\gamma^{-2} &= (1 - {\lvert \mathbf{v}/c \rvert}^2) \\

d\tau &= {\left(\frac{dx}{d\lambda} \cdot \frac{dx}{d\lambda}\right)}^{1/2} d\lambda \\

x \cdot x = {\lvert x \rvert}^2 &= c^2t^2 - {\lvert \mathbf{x} \rvert}^2 \\

E &= \int f \cdot (c d\tau) \\

\mathbf{v} &= \frac{d\mathbf{x}}{dt} \\

\mathbf{p} &= m\mathbf{v} \\

\end{align*}

[/tex]

Invarients for the first three four vectors are:

[tex]

\begin{align*}

{\lvert x \rvert}^2 &= c^2 t^2 - {\lvert \mathbf{x} \rvert}^2 = c^2 \tau^2 \\

{\lvert V \rvert}^2 &= \gamma^2 (c^2 - {\lvert \mathbf{v} \rvert}^2) = c^2 \\

{\lvert P \rvert}^2 &= m^2 {\lvert V \rvert}^2 = m^2 c^2 \\

\end{align*}

[/tex]

Is the minkowski norm of the four vector force:

[tex]

f = m\frac{d^2 x}{d\tau^2}

[/tex]

also an invarient? I think it has to be. Assuming that is the case, what would the value (and significance if any) of this be?

Last edited: