In appropriate relativity, we accept to change the announcement for beeline momentum.
Using m for blow mass, v and v for the object's acceleration and acceleration respectively, and c for the acceleration of ablaze in vacuum, we accept for beeline drive that \mathbf{p}=m\gamma \mathbf{v}, area \gamma = 1/\sqrt{1-v^2/c^2}.
Integrating by locations gives
E_k = \int \mathbf{v} \cdot d \mathbf{p}= \int \mathbf{v} \cdot d (m \gamma \mathbf{v}) = m \gamma \mathbf{v} \cdot \mathbf{v} - \int m \gamma \mathbf{v} \cdot d \mathbf{v} = m \gamma v^2 - \frac{m}{2} \int \gamma d (v^2)
Remembering that \gamma = (1 - v^2/c^2)^{-1/2}\!, we get:
\begin{align} E_k &= m \gamma v^2 - \frac{- m c^2}{2} \int \gamma d (1 - v^2/c^2) \\ &= m \gamma v^2 + m c^2 (1 - v^2/c^2)^{1/2} - E_0 \end{align}
where E0 serves as an affiliation constant. Thus:
\begin{align} E_k &= m \gamma (v^2 + c^2 (1 - v^2/c^2)) - E_0 \\ &= m \gamma (v^2 + c^2 - v^2) - E_0 \\ &= m \gamma c^2 - E_0 \end{align}
The connected of affiliation E0 is begin by celebratory that, if \mathbf{v }= 0 , \ \gamma = 1\! and E_k = 0 \!, giving
E_0 = m c^2 \,
and giving the accepted formula:
E_k = m \gamma c^2 - m c^2 = \frac{m c^2}{\sqrt{1 - v^2/c^2}} - m c^2
If a body's acceleration is a cogent atom of the acceleration of light, it is all-important to use relativistic mechanics (the approach of relativity as developed by Albert Einstein) to account its active energy.
For a relativistic article the drive p is according to:
p = \frac{m v}{\sqrt{1 - (v/c)^2}} .
Thus the plan expended accelerating an article from blow to a relativistic acceleration is:
E_k = \frac{m c^2}{\sqrt{1 - (v/c)^2}} - m c^2 .
The blueprint shows that the activity of an article approaches beyond as the acceleration v approaches the acceleration of ablaze c, appropriately it is absurd to advance an article beyond this boundary.
The algebraic by-product of this adding is the mass-energy adequation formula—the physique at blow accept to accept activity agreeable according to:
E_\text{rest} = E_0 = m c^2 \!
At a low acceleration (v<<="" p="">
E_k \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2\right) - m c^2 = \frac{1}{2} m v^2 ,
So, the absolute activity E can be abstracted into the activity of the blow accumulation additional the acceptable Newtonian active activity at low speeds.
When altar move at a acceleration abundant slower than ablaze (e.g. in accustomed phenomena on Earth), the aboriginal two agreement of the alternation predominate. The next appellation in the approximation is baby for low speeds, and can be begin by extending the amplification into a Taylor alternation by one added term:
E_k \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2 + \frac{3}{8} v^4/c^4\right) - m c^2 = \frac{1}{2} m v^2 + \frac{3}{8} m v^4/c^2 .
For example, for a acceleration of 10 km/s (22,000 mph) the alteration to the Newtonian active activity is 0.0417 J/kg (on a Newtonian active activity of 50 MJ/kg) and for a acceleration of 100 km/s it is 417 J/kg (on a Newtonian active activity of 5 GJ/kg), etc.
For college speeds, the blueprint for the relativistic active energy7 is acquired by artlessly adding the blow accumulation activity from the absolute energy:
E_k = m \gamma c^2 - m c^2 = m c^2\left(\frac{1}{\sqrt{1 - (v/c)^2}} - 1\right) .
The affiliation amid active activity and drive is added complicated in this case, and is accustomed by the equation:
E_k = \sqrt{p^2 c^2 + m^2 c^4} - m c^2.
This can aswell be broadcast as a Taylor series, the aboriginal appellation of which is the simple announcement from Newtonian mechanics.
What this suggests is that the formulas for activity and drive are not appropriate and axiomatic, but rather concepts which appear from the blueprint of accumulation with activity and the attempt of relativity.
edit Accepted relativity
See also: Schwarzschild geodesics
Using the assemblage that
g_{\alpha \beta} \, u^{\alpha} \, u^{\beta} \, = \, - c^2
where the four-velocity of a atom is
u^{\alpha} \, = \, \frac{d x^{\alpha}}{d \tau}
and \tau \, is the able time of the particle, there is aswell an announcement for the active activity of the atom in accepted relativity.
If the atom has momentum
p_{\beta} \, = \, m \, g_{\beta \alpha} \, u^{\alpha}
as it passes by an eyewitness with four-velocity uobs, again the announcement for absolute activity of the atom as empiric (measured in a bounded inertial frame) is
E \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta}
and the active activity can be bidding as the absolute activity bare the blow energy:
E_{k} \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} \, - \, m \, c^2 \, .
Consider the case of a metric which is askew and spatially isotropic (gtt,gss,gss,gss). Since
u^{\alpha} = \frac{d x^{\alpha}}{d t} \frac{d t}{d \tau} = v^{\alpha} u^{t} \,
where vα is the accustomed acceleration abstinent w.r.t. the alike system, we get
-c^2 = g_{\alpha \beta} u^{\alpha} u^{\beta} = g_{t t} (u^{t})^2 + g_{s s} v^2 (u^{t})^2 \,.
Solving for ut gives
u^{t} = c \sqrt{\frac{-1}{g_{t t} + g_{s s} v^2}} \,.
Thus for a anchored eyewitness (v= 0)
u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{t t}}} \,
and appropriately the active activity takes the form
E_k = - m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,.
Factoring out the blow activity gives:
E_k = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,.
This announcement reduces to the appropriate relativistic case for the flat-space metric where
g_{t t} = -c^2 \,
g_{s s} = 1 \,.
In the Newtonian approximation to accepted relativity
g_{t t} = - \left( c^2 + 2 \Phi \right) \,
g_{s s} = 1 - \frac{2 \Phi}{c^2} \,
where Φ is the Newtonian gravitational potential. This agency clocks run slower and barometer rods are beneath abreast massive bodies.
edit Active activity in breakthrough mechanics
Further information: Hamiltonian (quantum mechanics)
In breakthrough mechanics, observables like active activity are represented as operators. For one atom of accumulation m, the active activity abettor appears as a appellation in the Hamiltonian and is authentic in agreement of the added axiological drive abettor \hat p as
\hat T = \frac{\hat p^2}{2m}.
Notice that this can be acquired by replacing p by \hat p in the classical announcement for active activity in agreement of momentum,
E_k = \frac{p^2}{2m}.
In the Schrodinger picture, \hat p takes the anatomy -i\hbar\nabla area the acquired is taken with account to position coordinates and hence
\hat T = -\frac{\hbar^2}{2m}\nabla^2.
The apprehension amount of the electron active energy, \langle\hat{T}\rangle, for a arrangement of N electrons declared by the wavefunction \vert\psi\rangle is a sum of 1-electron abettor apprehension values:
\langle\hat{T}\rangle = \bigg\langle\psi \bigg\vert \sum_{i=1}^N \frac{-\hbar^2}{2 m_e} \nabla^2_i \bigg\vert \psi \bigg\rangle = -\frac{\hbar^2}{2 m_e} \sum_{i=1}^N \bigg\langle\psi \bigg\vert \nabla^2_i \bigg\vert \psi \bigg\rangle
where me is the accumulation of the electron and \nabla^2_i is the Laplacian abettor acting aloft the coordinates of the ith electron and the accretion runs over all electrons.
The body anatomic ceremonial of breakthrough mechanics requires ability of the electron body only, i.e., it formally does not crave ability of the wavefunction. Accustomed an electron body \rho(\mathbf{r}), the exact N-electron active activity anatomic is unknown; however, for the specific case of a 1-electron system, the active activity can be accounting as
T\rho = \frac{1}{8} \int \frac{ \nabla \rho(\mathbf{r}) \cdot \nabla \rho(\mathbf{r}) }{ \rho(\mathbf{r}) } d^3r
where Tρ is accepted as the von Weizsäcker active activity functional.
