Editing Spacetime (section)

=== Energy and momentum ===
{{Main|Four-momentum|Momentum|Mass–energy equivalence}}

==== Extending momentum to four dimensions ====
[[File:Relativistic spacetime momentum vector.svg|thumb|upright=1.5|Figure 3–8. Relativistic spacetime momentum vector. The coordinate axes of the rest frame are: momentum, p, and mass * c. For comparison, we have overlaid a spacetime coordinate system with axes: position, and time * c.]]
In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. [[Linear momentum]], the product of a particle's mass and velocity, is a [[Euclidean vector|vector]] quantity, possessing the same direction as the velocity: {{math|1='''''p'''''&nbsp;=&nbsp;''m'''v'''''}}. It is a ''conserved'' quantity, meaning that if a [[closed system]] is not affected by external forces, its total linear momentum cannot change.

In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector {{tmath|(x,t)}}. In exploring the properties of the spacetime momentum, we start, in Fig.&nbsp;3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. {{math|1=''p''&nbsp;=&nbsp;0}}, but the time component equals ''mc''.

We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that {{tmath|1=(m c)^{\prime}=\gamma m c}} and {{tmath|1=p^{\prime}=-\beta \gamma m c}}, since the red axes are rescaled by gamma. Fig.&nbsp;3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches ''c''.<ref name="Bais" />{{rp|84–87}}

We will use this information shortly to obtain an expression for the [[four-momentum]].

==== Momentum of light ====
[[File:Calculating the energy of light in different inertial frames.svg|thumb|Figure 3–9. Energy and momentum of light in different inertial frames]]
Light particles, or photons, travel at the speed of ''c'', the constant that is conventionally known as the ''speed of light''. This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a lightlike world line and, in appropriate units, have equal space and time components for every observer.

A consequence of [[Maxwell's theory]] of electromagnetism is that light carries energy and momentum, and that their ratio is a constant: {{tmath|1=E/p = c}}. Rearranging, {{tmath|1=E/c = p}}, and since for photons, the space and time components are equal, ''E''/''c'' must therefore be equated with the time component of the spacetime momentum vector.

Photons travel at the speed of light, yet have finite momentum and energy. For this to be so, the mass term in ''γmc'' must be zero, meaning that photons are [[massless particle]]s. Infinity times zero is an ill-defined quantity, but ''E''/''c'' is well-defined.

By this analysis, if the energy of a photon equals ''E'' in the rest frame, it equals {{tmath|1=E^{\prime}=(1-\beta) \gamma E }} in a moving frame. This result can be derived by inspection of Fig.&nbsp;3-9 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.<ref name="Bais" />{{rp|88}}

==== 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}}

==== Four-momentum ====
Because of the close relationship between mass and energy, the four-momentum (also called 4-momentum) is also called the energy–momentum 4-vector. Using an uppercase ''P'' to represent the four-momentum and a lowercase '''''p''''' to denote the spatial momentum, the four-momentum may be written as
: <math>P \equiv (E/c, \vec{p}) = (E/c, p_x, p_y, p_z)</math> or alternatively,
: <math>P \equiv (E, \vec{p}) = (E, p_x, p_y, p_z) </math> using the convention that <math>c = 1 .</math><ref name="Morin" />{{rp|129–130,180}}

{{anchor|Conservation laws}}