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Short answer: Mass and energy are NOT equivalent in general. Confusion can arise due to multiple definitions of mass (see our https://www.physicsforums.com/threads/what-is-relativistic-mass-and-why-it-is-not-used-much.796527/ ), but using the modern convention of identifying the word "mass" with the "invariant mass" (also known as "rest mass") it is clear that mass and energy are not equivalent.
Definition/Summary
##E = mc^2## is the famous formula relating mass to energy in the inertial reference frame where the mass is at rest. In SI units, ##E## is energy, in joules (J), and ##m## is mass, in kilograms (kg). Note, ##c^2## does not imply that the mass is moving at ##c##. Instead, ##c^2## reflects the fact that SI units are not natural units, so it is necessary to convert the units on the left to match the units on the right. In natural units where ##c=1## the formula would be simply ##E = m##.
Although this formula gives the impression that mass and energy are equivalent, this formula is itself a special case of a more general formula:
##E^2 = (mc^2)^2+(pc)^2##
The general equation reduces to the famous equation for ##p=0##. In other words, the common concept of mass energy equivalence holds only in the special case when ##p=0##. However, when ##p\ne 0## it is clear that mass and energy are not equivalent. In fact, when ##E=pc## you have energy, but no mass. This is the case for massless radiation, such as a pulse of light. Since ##p^2## can never be negative it is clear that all mass has energy, but the reverse is not true and it is possible to have energy without mass.
Although mass and energy are not equivalent in general, in an inertial frame where ##p=0## for some system the internal energy of the system is part of its mass. Since mass has inertia this can lead to interesting effects. For example, a box containing hot springs, or compressed springs, is more difficult to push than an identical box containing identical but cold or uncompressed springs. This becomes particularly important in nuclear physics and particle accelerators where more massive particles can be created from systems of smaller particles with high internal KE, and vice versa.
Definition/Summary
##E = mc^2## is the famous formula relating mass to energy in the inertial reference frame where the mass is at rest. In SI units, ##E## is energy, in joules (J), and ##m## is mass, in kilograms (kg). Note, ##c^2## does not imply that the mass is moving at ##c##. Instead, ##c^2## reflects the fact that SI units are not natural units, so it is necessary to convert the units on the left to match the units on the right. In natural units where ##c=1## the formula would be simply ##E = m##.
Although this formula gives the impression that mass and energy are equivalent, this formula is itself a special case of a more general formula:
##E^2 = (mc^2)^2+(pc)^2##
The general equation reduces to the famous equation for ##p=0##. In other words, the common concept of mass energy equivalence holds only in the special case when ##p=0##. However, when ##p\ne 0## it is clear that mass and energy are not equivalent. In fact, when ##E=pc## you have energy, but no mass. This is the case for massless radiation, such as a pulse of light. Since ##p^2## can never be negative it is clear that all mass has energy, but the reverse is not true and it is possible to have energy without mass.
Although mass and energy are not equivalent in general, in an inertial frame where ##p=0## for some system the internal energy of the system is part of its mass. Since mass has inertia this can lead to interesting effects. For example, a box containing hot springs, or compressed springs, is more difficult to push than an identical box containing identical but cold or uncompressed springs. This becomes particularly important in nuclear physics and particle accelerators where more massive particles can be created from systems of smaller particles with high internal KE, and vice versa.
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