Translation examples
Thus, the speed of "light" is also the speed of gravitational waves and any massless particle.
Valonnopeus on myös kaiken informaation, myös massattomien hiukkasten, nopeuden yläraja.
The square of the norm is: ‖ K ‖ 2 = K μ K μ = ( ω c ) 2 − k ⋅ k , {\displaystyle \|\mathbf {K} \|^{2}=K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-\mathbf {k} \cdot \mathbf {k} \,,} and by the de Broglie relation: ‖ K ‖ 2 = 1 ℏ 2 ‖ P ‖ 2 = ( m 0 c ℏ ) 2 , {\displaystyle \|\mathbf {K} \|^{2}={\frac {1}{\hbar ^{2}}}\|\mathbf {P} \|^{2}=\left({\frac {m_{0}c}{\hbar }}\right)^{2}\,,} we have the matter wave analogue of the energy–momentum relation: ( ω c ) 2 − k ⋅ k = ( m 0 c ℏ ) 2 . {\displaystyle \left({\frac {\omega }{c}}\right)^{2}-\mathbf {k} \cdot \mathbf {k} =\left({\frac {m_{0}c}{\hbar }}\right)^{2}\,.} Note that for massless particles, in which case m0 = 0, we have: ( ω c ) 2 = k ⋅ k , {\displaystyle \left({\frac {\omega }{c}}\right)^{2}=\mathbf {k} \cdot \mathbf {k} \,,} or ||k|| = ω/c.
Tämän normin neliö on: ‖ K ‖ 2 = K μ K μ = ( ω c ) 2 − k ⋅ k , {\displaystyle \|\mathbf {K} \|^{2}=K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-\mathbf {k} \cdot \mathbf {k} \,,} ja yhdistämällä tämä sekä de Broglien relaatio ‖ K ‖ 2 = 1 ℏ 2 ‖ P ‖ 2 = ( m c ℏ ) 2 , {\displaystyle \|\mathbf {K} \|^{2}={\frac {1}{\hbar ^{2}}}\|\mathbf {P} \|^{2}=\left({\frac {mc}{\hbar }}\right)^{2}\,,} saadaan energian ja liikemäärän yhteyttä vastaava relaatio aineaalloille: ( ω c ) 2 − k ⋅ k = ( m c ℏ ) 2 . {\displaystyle \left({\frac {\omega }{c}}\right)^{2}-\mathbf {k} \cdot \mathbf {k} =\left({\frac {mc}{\hbar }}\right)^{2}\,.} Voidaan todeta, että massattomilla hiukkasilla (m = 0) tästä saadaan: ( ω c ) 2 = k ⋅ k , {\displaystyle \left({\frac {\omega }{c}}\right)^{2}=\mathbf {k} \cdot \mathbf {k} \,,} tai ||k|| = ω/c.
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