Mass in special relativity

The term mass in special relativity can be used in different ways, occasionally leading to confusion. Historically, mass can refer to either the invariant mass or the relativistic mass.

  • The rest mass or invariant mass is an observer-independent quantity.
  • The relativistic mass or apparent mass depends on one's frame of reference.

In particular, the relativistic mass increases with observed velocity while the rest mass is an intrinsic property of an object, a so-called invariant.

In the earlier years of relativity, relativistic mass was taken to be the "correct" notion of mass, and the invariant mass was referred to as the rest mass. Gradually, with the development of Minkowski four-vector notation and general relativity, it was realized that the invariant mass was the more fundamental quantity in the theory of relativity.

The accepted usage in the scientific community today (at least in the context of special relativity) considers the invariant mass to be the only "mass", while the concept of energy has replaced the relativistic mass. In popular science and basic relativity courses, however, the relativistic mass is usually still presented, due to its conceptual simplicity and the fact that certain equations from nonrelativistic mechanics retain their form (namely, Newton's second law).

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The relativistic mass concept

According to the theory of relativity, an object with mass cannot travel at the speed of light. As such an object approaches the speed of light with respect to an observer, the observer will conclude that the object's kinetic energy is increasing toward infinity. Certain experiments (but not all) will also exhibit an increased inertia for the object associated with the increase in relativistic mass. The Lorentz gamma factor determines the observed increase in mass:

<math>\gamma = {1 \over {\sqrt{1 - v^2/c^2}}} \,\!</math>

When v, the velocity, is zero, <math>\gamma</math> is simply equal to 1, and the relativistic mass is reduced to the rest mass as you can see in the next two equations below. </br>As v increases toward the speed of light c, the denominator of the right side approaches zero, and consequently <math>\gamma</math> approaches infinity. The relativistic mass M is then formulated as:

<math>M = {m \over {\sqrt{1 - v^2/c^2}}} \,\!</math>

or

<math>M = \gamma{} m \,\!</math>

where m is the rest mass.

The main benefit of using the relativistic mass is that the formulas

<math>f=\frac{dp}{dt} \,\!</math>

and

<math>p=mv \,\!</math>

from nonrelativistic mechanics retain their form, and are valid for relativistic situations when used with M in place of m. The first equation is Newton's second law, the second is simply the definition of momentum.

Note, however, that many relations, e.g., Newton's second law in the form <math>\mathbf{F}=m\mathbf{a}</math>, do not work right if one simply replaces <math>m\,</math> by <math>\gamma m \,</math>. The correct relativistic form of <math>\mathbf{F}=m\mathbf{a}</math> is actually

<math>F_x = \gamma^3 m a_x \,</math>
<math>F_y = \gamma m a_y \,</math>
<math>F_z = \gamma m a_z \,</math>

(assuming that the velocity is along the <math>x</math> direction). For this reason, the use of the concept of relativistic mass is limited.

Another downside of this approach is that since <math>\gamma</math> depends on velocity, observers in different inertial reference frames will measure different values, which can be complicated. A more crucial flaw is that <math>\gamma</math> is undefined for v = c; in other words, these equations are not valid for photons.

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Kinetic energy

If M is the relativistic mass and m is the rest mass, with E being the total energy, we have:

<math>E = Mc^2 = m \gamma c^2 = {{mc^2} \over {\sqrt{1 - {{v^2} \over {c^2}}}}}</math>

The corresponding Taylor series is:

<math> E = mc^2+{{mv^2} \over 2}+{{3mv^4} \over {8c^2}}+{{5mv^6} \over {16c^4}}+\dots</math>

The first term (mc2) is the rest energy. The rest <math>\left({{mv^2} \over 2}+{{3mv^4} \over {8c^2}}+{{5mv^6} \over {16c^4}}+\dots\right)</math> is known as the kinetic energy. Except for speeds a sizable fraction of c, the terms with c in the denominator are negligible, therefore we obtain the commonly used formula for kinetic energy in Newton's system: <math>E_k = \begin{matrix} \frac{1}{2} \end{matrix}mv^2</math>.</br> Then it follows that for low velocities, <math>E \simeq mc^2 + \begin{matrix} \frac{1}{2} \end{matrix}mv^2</math>, which is the relativistic energy and not the Newtonian energy which consists uniquely of the kinetic energy.

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The relativistic energy-momentum equation

The relativistic expressions for E and p above can be manipulated into the fundamental relativistic energy-momentum equation:

<math>E^2 - (pc)^2 = (mc^2)^2 \,\!</math>

Note that there is no relativistic mass in this equation. The equation is also valid for photons, which are massless:

<math>E^2 - (pc)^2 = 0 \,\!</math>
<math>E^2 = (pc)^2 \,\!</math>
<math>E = pc \,\!</math>
<math>p = E/c \,\!</math>

Therefore a photon's momentum is a function of its energy; it is not analogous to the momentum in Newtonian mechanics.

Considering an object at rest, the momentum p, in the first equation above, is zero, and we obtain

<math>E^2 = (mc^2)^2 \,\!</math>

which reduces to

<math>E = mc^2 \,\!</math>

suggesting that this last well-known relation is only valid when the object is at rest, giving what is known as the rest energy. If the object is in motion, we have

<math>E^2 = (mc^2)^2 + (pc)^2 \,\!</math>

From this we see that the total energy of the object E depends on its rest energy and momentum; as the momentum increases with the increase of the velocity v, so does the total energy.

This E is in fact equivalent to that of the relativistic energy equation in the previous section, and that energy equation differs from the relativistic mass equation by a factor of c2. Therefore the relativistic mass is essentially the same as the total energy — but scaled and with different units. Since the energy-momentum equation is more convenient to use (especially with four-vector notation), the relativistic mass is never used in practice.

When working in units where c = 1, known as the natural unit system, the energy-momentum equation reduces to

<math>E^2 - p^2 = m^2 \,\!</math></br>

which itself reduces to E² = m² or E = m when v = 0. Energy is typically in units of electron volts (eV), momentum in units of eV/c, and mass in units of eV/c2. This is the primary unit system in particle physics.</br> Energy can also be expressed in units of grams, as for the mass.

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