Newton's second law, $\vec{F}=m\vec{a}$, is form-invariant only under Galilean transformations but not under Lorentz transformations. Then why do we say that Newton's law is valid and form-invariant in any inertial frame?
Is the definition of inertial frame different in Newtonian physics and Special relativity? So that when we say Newton's law is valid in any inertial frame we mean inertial frame defined as per Newton's first law?
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asked Aug 14, 2020 at 14:05
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$\begingroup$ Galilean transformation assumes the possibility to have infinitely large speed of an object. It assumes no connection between time and space, whereas Lorentz transform couples time-space introducing 4D topology. Surely, the definition of invariance and inertia (mass) differs, as well as the definition of a fundamental force. I would suggest to refresh the axiomatics of both theories in parallel and compare them. $\endgroup$– MsTaisCommented Aug 14, 2020 at 14:34
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3 Answers
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Are you sure we say such a think that Newton's second law is form invariant in any inertial frame? I think both of your comments are correct except of this claim.
Newton's second law not only changes under Lorentz transforming from a frame to another, but it is not even correct in one frame when applied to moving objects. Since the correct law would read: $$\vec{F} = \gamma m\vec{a}_\bot + \gamma^3 m\vec{a}_\parallel$$
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$\begingroup$ It depends on what you mean by an inertial frame. Within the realm of Newtonian mechanics, inertial frames are defined as those frames in which Newton's first law is valid i.e., an object subjected to no net physical forces will remain unaccelerated. You're writing the modified version of Newton's law in special relativity. If you define an inertial frame like this, then all frames connected to it by a Galilean transformation are also internal. Under Galilean transformation, therefore, $\vec{F}=m\vec{a}$ is form-invariant. $\endgroup$ Commented Aug 14, 2020 at 14:44
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$\begingroup$ Also, could you give the formulas for $\vec{a}_\perp$ and $\vec{a}_{||}$? $\endgroup$ Commented Aug 14, 2020 at 14:50
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$\begingroup$ I think your comment, in return, is assuming that the Newton's second law is defined as the equation I wrote for the force. But this is not the second law. Of course if you change the definition of the second law to what I wrote, then it becomes valid in Lorentz transformed frames, but that is how I derived the equation for force; that is by assuming the invariance. Also $\vec{a}$ is defined as it is usually defined, with respect to the coordinates of the current frame. $\endgroup$– AlphyCommented Aug 14, 2020 at 17:14
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It actually is possible to write Newtons second law in a covariant(=forminvariant under Lorentz transformations) form.
When generalizing to relativistic particle mechanics, it is important to keep track of the reference systems where you define your quantities in. The trick is to derive with respect to the proper time instead of the time of your preferred frame. Then you can write the covariant equation $$F^\mu = m\frac{d^2 x^\mu}{d\tau^2}$$ with proper time $\tau$, position 4-vector $x^\mu$ and a relativistic force $F^\mu$ defined by $F^\mu = (0,\vec{F})$ in the rest frame of the particle.
See S. Weinberg, Gravitation and Cosmology, ch. 2.3 for a more detailed explanation.
EDIT: The difference between Newton mechanics and special relativity is that there is a finite maximal velocity $c$ in special relativity(one of Einsteins postulates). This leads to the necessity of using Lorentz transformations instead of Galilei transformations to describe changes between inertial frames properly. Since one defines Galilei/Lorentz transformations to be the transformations that change between inertial frames, the definition of inertial frame also changes when one has a finite $c$. Note that one gets Newtonian mechanics from special relativity in the limit $c\to\infty$.
To answer the question: Newtons second law is invariant under Galilei transformations for $c=\infty$. Since one finds $c<\infty$ in experiments, one has to use Lorentz transformations and the form of Newtons second law discussed above is invariant under Lorentz transformations. However, one usually does not work with velocities where the difference between $c=\infty$/$c<\infty$ or Galilei/Lorentz transformation is important, so Newtonian mechanics is still a very good approximation in most cases, even though it is not exact for high velocities as special relativity.
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$\begingroup$ Within the realm of Newtonian mechanics, inertial frames are defined as those frames in which Newton's first law is valid i.e., an object subjected to no net physical forces will remain unaccelerated. How do you define an inertial frame in special relativity? $\endgroup$ Commented Aug 14, 2020 at 14:48
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$\begingroup$ After studying Wikipedia, I think I answered your question properly. In special relativity, one first defines Lorentz transformations as linear transformations on a space with metric $(1,-1,-1,-1)$. Then, one defines inertial frames as the things that a Lorentz transformations of an inertial frame transforms into. $\endgroup$– jonasCommented Aug 14, 2020 at 15:52
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The definition of an inertial reference frame is the same. What is different is that special relativity takes for granted a physical fact which Newton did not have any reason to suspect.
This physical fact is that, when you see a clock at a coördinate $x$ along some axis along which you are accelerating with some acceleration $\alpha$, then once you correct for the Doppler shift it appears to tick faster at a rate $1+\alpha x/c^2$ seconds per second. If $x$ is behind you along this axis then it will appear to tick slower. When you cease accelerating comet to first order approximation it will return to ticking at the same rate but it will have some $x$-dependent offset and there are second order effects which will mean that it actually appears to tick a little slowly.
One can view this as defining a characteristic length scale for a given acceleration $L=c^2/\alpha$, over which you begin to notice gravitational effects. Before special relativity came along, people mostly had experience with accelerations on the order of 10 m/s² or so, and the characteristic length scale for this is one light-year. Since nobody had interstellar interactions, nobody would have noticed large applications of this effect. Or to put it another way, if you’re doing experiments that require line-of-sight on earth’s surface you are going to be limited in distance to laboratories that are maybe 10km apart or so, maybe on opposite sides of a valley. Meanwhile a clock in the mid 1700s was good if it only lost a second per day and quartz resonators only improved this to half a second per day, so a tenth seems generous: at that distance and accuracy, you would need to create accelerations in the lab of approximately a million g's.
It required the development of highly sensitive measurement apparatus for us to suspect and then confirm it, as well as a wildly promising theory of electromagnetism which transformed under the Lorentz group instead of the Galilean group to nudge us that last little bit. In particular, one consequence of this way that acceleration causes clocks to tick, is that everybody sees light move at the same speed, and that is what the Michelson-Morley interferometer eventually confirmed.
