Summarizing Relativistic Energy

Relativistic Energy Summary

• Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
• Total Energy is defined as: \(E={\mathrm{\gamma mc}}^{2}\), where \(\gamma =\cfrac{1}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}\).
• Rest energy is \({E}_{0}={\mathrm{mc}}^{2}\), meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.
• We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.
• The relativistic work-energy theorem is \({W}_{\text{net}}=E-{E}_{0}=\gamma {\mathrm{mc}}^{2}-{\mathrm{mc}}^{2}=(\gamma -1){\mathrm{mc}}^{2}\).
• Relativistically, \({W}_{\text{net}}={\text{KE}}_{\text{rel}}\) , where \({\text{KE}}_{\text{rel}}\) is the relativistic kinetic energy.
• Relativistic kinetic energy is \({\text{KE}}_{\text{rel}}=(\gamma -1){\mathrm{mc}}^{2}\), where \(\gamma =\cfrac{1}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}\). At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
• No object with mass can attain the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
• The equation \({E}^{2}=(\mathrm{pc}{)}^{2}+({\mathrm{mc}}^{2}{)}^{2}\) relates the relativistic total energy \(E\) and the relativistic momentum \(p\). At extremely high velocities, the rest energy \({\mathrm{mc}}^{2}\) becomes negligible, and \(E=\mathrm{pc}\).

Glossary

total energy

defined as \(E={\mathrm{\gamma mc}}^{2}\), where \(\gamma =\cfrac{1}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}\)

rest energy

the energy stored in an object at rest: \({E}_{0}={\mathrm{mc}}^{2}\)

relativistic kinetic energy

the kinetic energy of an object moving at relativistic speeds: \({\text{KE}}_{\text{rel}}=(\gamma -1){\mathrm{mc}}^{2}\), where \(\gamma =\cfrac{1}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}\)