Low-speed expansion

Using the Lorentz factor, γ, the energy–momentum can be rewritten as E = γmc2 and expanded as a power series:

${\displaystyle E=m_0{c}^{2}1+\frac{1}{2}{\left(\frac{v}{c}\right)}^2+\frac{3}{8}{\left(\frac{v}{c}\right)}^4+\frac{5}{16}{\left(\frac{v}{c}\right)}^6+\dots <!--\; \dots \; -->.}$

For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because v/c is small. For low speeds, all but the first two terms can be ignored:

${\displaystyle E\approx <!--\; \approx \; -->m_0{c}^2+\frac{1}{2}{m}_0{v}^2.}$

In classical mechanics, both the m₀c2 term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured, so the m₀c2 term is ignored in classical physics. While the higher-order terms become important at higher speeds, the Newtonian equation is a highly accurate low-speed approximation; adding in the third term yields:

${\displaystyle E\approx <!--\; \approx \; -->m_0{c}^2+\frac{1}{2}{m}_0{v}^2\left(1+\frac{3v^2}{4c^2}\right)}$.

The difference between the two approximations is given by ${\displaystyle \frac{3v^2}{4c^2}}$, a number very small for everyday objects. In 2018 NASA announced the Parker Solar Probe was the fastest ever, with a speed of 153,454 miles per hour (68,600 m/s). The difference between the approximations for the Parker Solar Probe in 2018 is ${\displaystyle \frac{3v^2}{4c^2}\approx <!--\; \approx \; -->3.9\ast <!--\; \ast \; -->{10}^{-<!--\; -\; -->8}}$, which accounts for an energy correction of four parts per hundred million. The gravitational constant, in contrast, has a standard relative uncertainty of about ${\displaystyle 2.2\ast <!--\; \ast \; -->{10}^{-<!--\; -\; -->5}}$.