No external force of scenario A

With external force of scenario B

Δm

$\left(1-\sqrt{1-\frac{{\nu }^2}{c^2}}\right)m_0$

$\left(\frac{1}{\sqrt{1-\frac{{\nu }^2}{c^2}}}-1\right)m_0$

Origin of Δm

internal

external

Net kinetic energy $\Delta E_k$

$\left(1-\sqrt{1-\frac{{\nu }^2}{c^2}}\right)m_0{c}^2$

$\left(\frac{1}{\sqrt{1-\frac{{\nu }^2}{c^2}}}-1\right)m_0{c}^2$

Intrinsic kinetic energy $E_k=\Delta E_k+E_{k-rest}$

$\left(2-\sqrt{1-\frac{{\nu }^2}{c^2}}\right)m_0{c}^2$

$\frac{m_0{c}^2}{\sqrt{1-\frac{{\nu }^2}{c^2}}}$

Intrinsic potential energy $E_s=E_0-E_k$

$\left(\sqrt{1-\frac{{\nu }^2}{c^2}}\right)m_0{c}^2$

$m_0{c}^2$

Δm when $v=c$

$\Delta m=m_0$

$\Delta m=\infty$

Mass m' when $v<c$

$m^{\prime }=m_0\sqrt{1-\frac{{\nu }^2}{c^2}}$

$m^{\prime }=\frac{m_0}{\sqrt{1-\frac{{\nu }^2}{c^2}}}$

Lorentz Invariant $m_0^2{c}^4$

$m_0^2{c}^4={m^{\prime }}^2{c}^4+{\left(m_{0}v\right)}^2{c}^2$

$m_0^2{c}^4={m^{\prime }}^2{c}^4-p^2{c}^2$

Mass m' changes when v increases

decrease

increase

Example

Gravity

External force with applying work