Editing Spacetime (section)

==== Mass–energy relationship ====
Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several important conclusions.

*In the low speed limit as {{math|1=''β'' = ''v''/''c''}} approaches zero, {{mvar|&gamma;}} approaches 1, so the spatial component of the relativistic momentum {{tmath|1=\beta \gamma m c=\gamma m v}} approaches ''mv'', the classical term for momentum. Following this perspective, ''&gamma;m'' can be interpreted as a relativistic generalization of ''m''. Einstein proposed that the ''[[relativistic mass]]'' of an object increases with velocity according to the formula {{tmath|1=m_\text{rel}=\gamma m}}.
*Likewise, comparing the time component of the relativistic momentum with that of the photon, {{tmath|1=\gamma m c=m_\text{rel} c=E / c}}, so that Einstein arrived at the relationship {{tmath|1=E=m_\text{rel} c^{2} }}. Simplified to the case of zero velocity, this is Einstein's equation relating energy and mass.

Another way of looking at the relationship between mass and energy is to consider a series expansion of {{math|1=''&gamma;mc''<sup>2</sup>}} at low velocity:
: <math> E = \gamma m c^2 =\frac{m c^2}{\sqrt{1 - \beta ^ 2}}</math> <math>\approx m c^2 + \frac{1}{2} m v^2 ...</math>
The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy.<ref name="Bais" />{{rp|90–92}}<ref name="Morin" />{{rp|129–130,180}}

The concept of relativistic mass that Einstein introduced in 1905, ''m''<sub>rel</sub>, although amply validated every day in particle accelerators around the globe (or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes,<ref>{{cite journal|last1=Rose|first1=H. H.|title=Optics of high-performance electron microscopes|journal=Science and Technology of Advanced Materials|date=21 April 2008|volume=9|issue=1|page=014107|doi=10.1088/0031-8949/9/1/014107|bibcode=2008STAdM...9a4107R|pmc=5099802|pmid=27877933}}</ref> old-fashioned color television sets, etc.), has nevertheless not proven to be a ''fruitful'' concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity.

For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy.<ref>{{cite book |last1=Griffiths |first1=David J. |title=Revolutions in Twentieth-Century Physics |date=2013 |publisher=Cambridge University Press |location=Cambridge |isbn=978-1-107-60217-5 |page=60 |url=https://books.google.com/books?id=Tv8cz-kN2z0C&pg=PA60 |access-date=24 May 2017 |language=en}}</ref> "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or [[invariant mass]], and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula,
: <math> E^2 - p^2c^2 = m_\text{rest}^2 c^4 </math>
This formula applies to all particles, massless as well as massive. For photons where ''m''<sub>rest</sub> equals zero, it yields, {{tmath|1=E=\pm p c}}.<ref name="Bais" />{{rp|90–92}}