Basic question about inertial reference frame

[sunjin09]

I have no background in relativity. Recently I started reading some introduction to special relativity in Griffith's EM book, where he vaguely defined an inertial reference frame as one in which Newton's first law holds. Now according to this definition, does such frame exist in nature?

On the other hand, assuming such frame exists, does this make special relativity self-consistent along with Maxwell's equations and some "enhanced" version of Newton's mechanics, or is it inevitable that some contradictions demand further generalization leading to e.g., general relativity?

 

[Dale]

Reference frames of any kind don't "exist in nature". They are man made mathematical constructs. They were invented, not discovered.

 

[PeterDonis]

I have no background in relativity. Recently I started reading some introduction to special relativity in Griffith's EM book, where he vaguely defined an inertial reference frame as one in which Newton's first law holds. Now according to this definition, does such frame exist in nature?

Does a frame exist in which Newton's First Law holds *exactly*? No. But frames do exist in which Newton's First Law holds to a very, very good approximation. The smaller the range in space and time that you need the frame to cover, and the further away the frame is from gravitating bodies, the better the approximation.

On the other hand, assuming such frame exists, does this make special relativity self-consistent along with Maxwell's equations and some "enhanced" version of Newton's mechanics, or is it inevitable that some contradictions demand further generalization leading to e.g., general relativity?

The definition takes care of Newton's First Law. As long as gravity is negligible, you can come up with versions of Newton's Second and Third Laws that are consistent with SR. But to incorporate gravity, you need GR.

 

[sunjin09]

Reference frames of any kind don't "exist in nature". They are man made mathematical constructs. They were invented, not discovered.

Thank you, that answers my first question and leads to the second one, i.e., if I define a inertial reference frame in which Maxwell equations as well as SR hold and ignore gravity. Is this "mathematical" framework completely self-consistent? Do we need to modify Newton's second and third law?

 

[sunjin09]

The definition takes care of Newton's First Law. As long as gravity is negligible, you can come up with versions of Newton's Second and Third Laws that are consistent with SR. But to incorporate gravity, you need GR.

Thank you. Are Maxwell's equations as well as Lorentz force law completely compatible, assuming there's no "point charge", only continuous distribution of charges exists?

 

[PeterDonis]

Thank you. Are Maxwell's equations as well as Lorentz force law completely compatible, assuming there's no "point charge", only continuous distribution of charges exists?

If you mean completely compatible with SR, yes.

 

[PeterDonis]

Reference frames of any kind don't "exist in nature". They are man made mathematical constructs. They were invented, not discovered.

In the light of this (valid) comment, I should clarify that I was interpreting "reference frame" to mean "state of motion of an observer". With this interpretation the OP's question would be rephrased as "can observers in nature be in states of motion such that they observe Newton's First Law to hold?"

 

[sunjin09]

If you mean completely compatible with SR, yes.

Griffiths also mentioned another paradox earlier in his book regarding a point charge's radiation reaction, where he derived a dumbbell shaped accelerating charge having infinite mass, and he mentioned in the footnote that such paradox is "covered up" under quantum EM. His derivation of that paradox used Newton's second law. Now if I assume there's no point charge, only continuous distribution of charge, as well as making necessary changes to Newton's laws, can I make everything compatible under a "closed" mathematical system?

 

[m4r35n357]

Is the phrase "inertial reference frame" fundamentally confusing to beginners? Just as an example, in Susskind's GR video series he talks of a circular argument something like "in an inertial reference frame motion is uniform" vs "an inertial frame is defined as one in which motion is uniform".
I can see the sense in an inertial _observer_, ie a weightless observer, but a whole frame of reference? Also why is the word "weightless" used so little, at least in introductory texts, to me it seems absolutely central to understanding inertial motion?
Perhaps an inertial frame is just a "field of weightless observers"?
Hope these questions make sense!

 

[PeterDonis]

His derivation of that paradox used Newton's second law.

I don't have Griffiths' book, so I can't check directly, but I would assume that his derivation of the paradox also used an infinite charge density at some point in space, since that's what "point charge" means. It's the infinite charge density that creates the problem, not Newton's second law.

Now if I assume there's no point charge, only continuous distribution of charge, as well as making necessary changes to Newton's laws, can I make everything compatible under a "closed" mathematical system?

Yes.

 

[PeterDonis]

Perhaps an inertial frame is just a "field of weightless observers"?

Basically, yes, though the observers don't have to actually exist. More precisely, it's a field of weightless observers who are all at rest relative to one another (and who can continuously verify this by exchanging light signals and measuring constant round-trip travel times).

If there is no gravity, such a field of observers can cover the entire spacetime. However, if gravity is present, then weightless observers who are at rest relative to each other at some instant will not remain at rest; they will find themselves moving apart or moving together due to the effects of gravity (more precisely, tidal gravity). So such a field of observers can only cover a small portion of spacetime (i.e., a small extent in space and a small interval of time). How small depends on how accurately the observers can measure the relative motion induced by tidal gravity.

 

[Dale]

Thank you, that answers my first question and leads to the second one, i.e., if I define a inertial reference frame in which Maxwell equations as well as SR hold and ignore gravity. Is this "mathematical" framework completely self-consistent?

Yes.

Do we need to modify Newton's second and third law?

You need to express them in a form which is consistent with special relativity, but you do not need to add any fictitious forces like you do in a non inertial frame.

 

[m4r35n357]

Basically, yes, though the observers don't have to actually exist. More precisely, it's a field of weightless observers who are all at rest relative to one another (and who can continuously verify this by exchanging light signals and measuring constant round-trip travel times).

Ah, good point, I hadn't considered the need for them to be at rest wrt each other ;) In that case I think I ought to change my "definition" to "a frame constructed around a weightless observer". However, I still think the word "weightless" conveys more meaning than "inertial" to a newcomer to the subject (unless it is incorrect, of course!).