What do the small terms in the series expansion of relativistic energy mean?

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Through the fabulous Feynman Lectures of Physics and the introduction of relativistic mass, Richard Feynman made a link between the increase in kinetic energy of a heated molecule of gas, and its relativistic increase of mass.



The explanation presented here (http://www.feynmanlectures.caltech.edu/I_15.html section 15.8) takes the expression of the mass as a function of velocity, according to Einstein's relativity:
$$
m = m_0 frac1sqrt1-fracv^2c^2 = m_0 Big(1-fracv^2c^2Big)^-1/2
$$

He then expands it in a power series, which gives the following terms:
$m_0$ which is the mass at rest, $frac12 m_0 v^2 Big(frac1c^2Big)$ which is the apparent increase of mass caused by velocity (kinetic energy divided by $c^2$), and then... negligible terms, which makes sense due to $c^-4$, $c^-6$, etc.



Mathematics being of a great help in such precision physics, I wondered whether the following terms would actually physically represent something, like some sort of other velocity-dependent energy. Actually, due to the small proportion of these terms at low velocities, this would be non-negligible only at very high velocities, like particles very close to the celerity of speed.



My question is about the way to interpret the formula. Should we associate some other energy to $frac38 c^4 m_0 v^4$ and the following terms (but still related to speed...)? Or do we have to consider that the simple formula for kinetic energy $E_k = frac12 m_0 v^2$ was a good approximation with small speeds, but is not totally correct in the relativistic world and shall take also the following terms?










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    up vote
    2
    down vote

    favorite












    Through the fabulous Feynman Lectures of Physics and the introduction of relativistic mass, Richard Feynman made a link between the increase in kinetic energy of a heated molecule of gas, and its relativistic increase of mass.



    The explanation presented here (http://www.feynmanlectures.caltech.edu/I_15.html section 15.8) takes the expression of the mass as a function of velocity, according to Einstein's relativity:
    $$
    m = m_0 frac1sqrt1-fracv^2c^2 = m_0 Big(1-fracv^2c^2Big)^-1/2
    $$

    He then expands it in a power series, which gives the following terms:
    $m_0$ which is the mass at rest, $frac12 m_0 v^2 Big(frac1c^2Big)$ which is the apparent increase of mass caused by velocity (kinetic energy divided by $c^2$), and then... negligible terms, which makes sense due to $c^-4$, $c^-6$, etc.



    Mathematics being of a great help in such precision physics, I wondered whether the following terms would actually physically represent something, like some sort of other velocity-dependent energy. Actually, due to the small proportion of these terms at low velocities, this would be non-negligible only at very high velocities, like particles very close to the celerity of speed.



    My question is about the way to interpret the formula. Should we associate some other energy to $frac38 c^4 m_0 v^4$ and the following terms (but still related to speed...)? Or do we have to consider that the simple formula for kinetic energy $E_k = frac12 m_0 v^2$ was a good approximation with small speeds, but is not totally correct in the relativistic world and shall take also the following terms?










    share|cite|improve this question

























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Through the fabulous Feynman Lectures of Physics and the introduction of relativistic mass, Richard Feynman made a link between the increase in kinetic energy of a heated molecule of gas, and its relativistic increase of mass.



      The explanation presented here (http://www.feynmanlectures.caltech.edu/I_15.html section 15.8) takes the expression of the mass as a function of velocity, according to Einstein's relativity:
      $$
      m = m_0 frac1sqrt1-fracv^2c^2 = m_0 Big(1-fracv^2c^2Big)^-1/2
      $$

      He then expands it in a power series, which gives the following terms:
      $m_0$ which is the mass at rest, $frac12 m_0 v^2 Big(frac1c^2Big)$ which is the apparent increase of mass caused by velocity (kinetic energy divided by $c^2$), and then... negligible terms, which makes sense due to $c^-4$, $c^-6$, etc.



      Mathematics being of a great help in such precision physics, I wondered whether the following terms would actually physically represent something, like some sort of other velocity-dependent energy. Actually, due to the small proportion of these terms at low velocities, this would be non-negligible only at very high velocities, like particles very close to the celerity of speed.



      My question is about the way to interpret the formula. Should we associate some other energy to $frac38 c^4 m_0 v^4$ and the following terms (but still related to speed...)? Or do we have to consider that the simple formula for kinetic energy $E_k = frac12 m_0 v^2$ was a good approximation with small speeds, but is not totally correct in the relativistic world and shall take also the following terms?










      share|cite|improve this question















      Through the fabulous Feynman Lectures of Physics and the introduction of relativistic mass, Richard Feynman made a link between the increase in kinetic energy of a heated molecule of gas, and its relativistic increase of mass.



      The explanation presented here (http://www.feynmanlectures.caltech.edu/I_15.html section 15.8) takes the expression of the mass as a function of velocity, according to Einstein's relativity:
      $$
      m = m_0 frac1sqrt1-fracv^2c^2 = m_0 Big(1-fracv^2c^2Big)^-1/2
      $$

      He then expands it in a power series, which gives the following terms:
      $m_0$ which is the mass at rest, $frac12 m_0 v^2 Big(frac1c^2Big)$ which is the apparent increase of mass caused by velocity (kinetic energy divided by $c^2$), and then... negligible terms, which makes sense due to $c^-4$, $c^-6$, etc.



      Mathematics being of a great help in such precision physics, I wondered whether the following terms would actually physically represent something, like some sort of other velocity-dependent energy. Actually, due to the small proportion of these terms at low velocities, this would be non-negligible only at very high velocities, like particles very close to the celerity of speed.



      My question is about the way to interpret the formula. Should we associate some other energy to $frac38 c^4 m_0 v^4$ and the following terms (but still related to speed...)? Or do we have to consider that the simple formula for kinetic energy $E_k = frac12 m_0 v^2$ was a good approximation with small speeds, but is not totally correct in the relativistic world and shall take also the following terms?







      special-relativity mass-energy approximations






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      edited 20 mins ago









      David Z♦

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      asked 6 hours ago









      Teuxe

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          The formula $E_k=frac12m_0v^2$ is an accurate approximation for small values of $v/c$, but it begins to diverge from the correct value as $v$ becomes an appreciable fraction of $c$. The higher order terms you speak of simply measure the extent of the error, which is small when $vll c$ but becomes important as $vrightarrow c$.



          enter image description here



          Put a different way, the terms you're asking about aren't a different kind of energy - they're just corrections to the kinetic energy which become important at high speeds.






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            1 Answer
            1






            active

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            oldest

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            active

            oldest

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            up vote
            3
            down vote



            accepted










            The formula $E_k=frac12m_0v^2$ is an accurate approximation for small values of $v/c$, but it begins to diverge from the correct value as $v$ becomes an appreciable fraction of $c$. The higher order terms you speak of simply measure the extent of the error, which is small when $vll c$ but becomes important as $vrightarrow c$.



            enter image description here



            Put a different way, the terms you're asking about aren't a different kind of energy - they're just corrections to the kinetic energy which become important at high speeds.






            share|cite|improve this answer


























              up vote
              3
              down vote



              accepted










              The formula $E_k=frac12m_0v^2$ is an accurate approximation for small values of $v/c$, but it begins to diverge from the correct value as $v$ becomes an appreciable fraction of $c$. The higher order terms you speak of simply measure the extent of the error, which is small when $vll c$ but becomes important as $vrightarrow c$.



              enter image description here



              Put a different way, the terms you're asking about aren't a different kind of energy - they're just corrections to the kinetic energy which become important at high speeds.






              share|cite|improve this answer
























                up vote
                3
                down vote



                accepted







                up vote
                3
                down vote



                accepted






                The formula $E_k=frac12m_0v^2$ is an accurate approximation for small values of $v/c$, but it begins to diverge from the correct value as $v$ becomes an appreciable fraction of $c$. The higher order terms you speak of simply measure the extent of the error, which is small when $vll c$ but becomes important as $vrightarrow c$.



                enter image description here



                Put a different way, the terms you're asking about aren't a different kind of energy - they're just corrections to the kinetic energy which become important at high speeds.






                share|cite|improve this answer














                The formula $E_k=frac12m_0v^2$ is an accurate approximation for small values of $v/c$, but it begins to diverge from the correct value as $v$ becomes an appreciable fraction of $c$. The higher order terms you speak of simply measure the extent of the error, which is small when $vll c$ but becomes important as $vrightarrow c$.



                enter image description here



                Put a different way, the terms you're asking about aren't a different kind of energy - they're just corrections to the kinetic energy which become important at high speeds.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 33 mins ago

























                answered 5 hours ago









                J. Murray

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