Newton's law under Lorentz transformation

[davidge]

According to this pdf http://www.springer.com/cda/content/document/cda_downloaddocument/9783319011066-c2.pdf?SGWID=0-0-45-1429331-p175291974 Newton's second law is not invariant under Lorentz transformations. To find out the part that says so, use CTRL+F and type "Newton"; it's the first result that should appear on the search.

This pdf http://www.pa.uky.edu/~kwng/phy361/class/class5.pdf says in page 2 that both Maxwell's and Newton's equations are invariant under Lorentz transformations.

What is wrong?

 

[A Lazy Shisno]

The first one is correct. Newton's second law is covariant as it is a vector law that doesn't change its configuration when switching reference frames. Only scalars can be invariant in different frames, vectors can, at best, be covariant.

 

[davidge]

This sounds strange, because acceleration is a vector quantity, so between two inertial frames it seems like $d^2x / dt^2 = d^2 x' / dt'^2$. Knowing that the inertial mass is also invariant between two inertial frames, then Newton's second law should be invariant even under Lorentz transformations...

The very definition of a inertial frame as one in which the Newton's laws holds, should imply the invariance of Newton's equation.

 

[PeterDonis]

acceleration is a vector quantity

It's a 3-vector, not a 4-vector, at least not the way you're defining it. (There is a way to define a 4-acceleration vector, but you're not using it.)

between two inertial frames it seems like $d^2x / dt^2 = d^2 x' / dt'^2$.

Instead of just guessing what it "seems like", try actually doing a Lorentz transformation to check.

 

[PeterDonis]

Newton's second law is covariant as it is a vector law that doesn't change its configuration when switching reference frames

This needs to be stated very carefully. 3-vector laws are not necessarily even covariant. 4-vector laws are, but Newton's second law in its usual form can't be written as a 4-vector law in the general case.

 

[davidge]

@PeterDonis , how can it be that Newton's laws are invariant between inertial frames if the Newton's equation itself is not invariant?

 

[PeterDonis]

how can it be that Newton's laws are invariant between inertial frames

Did I say they were? (I assume you mean "invariant under Lorentz transformations".)

 

[davidge]

Did I say they were?

No. I'm assuming they are, because of the very definition of a inertial frame as one in which Newton's laws are valid.

 

[PeterDonis]

the very definition of a inertial frame as one in which Newton's laws are valid.

That's not the definition of an inertial frame. The definition of an inertial frame is one in which Newton's first law is valid--i.e., in which an object not subject to any forces moves in a straight line in a constant speed. There is no requirement that Newton's second law has to be valid in an inertial frame--at least not in its usual 3-vector form.

 

[davidge]

Ah, ok. But is there a way for the "Laws of nature to be the same in all inertial frames" (Postulate from General Relativity) with the equation of Newton not keeping the same form?

 

[DrGreg]

Ah, ok. But is there a way for the "Laws of nature to be the same in all inertial frames" (Postulate from General Relativity) with the equation of Newton not keeping the same form?

Newton's second law expressed in either of the 3-vector forms$$
F^i = m \frac{{\rm d}^2 x^i}{{\rm d}t^2}
$$(if $m$ doesn't change over time), or, more generally,$$
F^i = \frac{\rm d}{{\rm d}t} \left( m \frac{{\rm d}x^i}{{\rm d}t} \right)
$$isn't even correct in relativity, never mind invariant (because relativistic momentum isn't $m \, {\rm d}x^i / {\rm d}t$). The correct versions are the 4-vector equations$$
F^\alpha = m \frac{{\rm D}^2 x^\alpha}{{\rm d}\tau^2}
$$or$$
F^\alpha = \frac{\rm D}{{\rm d}\tau} \left( m \frac{{\rm d}x^\alpha}{{\rm d}\tau} \right)
$$

 

[davidge]

@DrGreg Thanks. It's surprising see how it looks the same as the incorrect one. It seems that the only difference is that time is accounted as one of the directions, i.e. the force becomes a Minkowski-four-vector. Did I get this right? Why can't one use this lorentz invariant form in Newton's theory?

 

[Nugatory]

Why can't one use this lorentz invariant form in Newton's theory?

If you do, you don't have Newton's theory. In Newton's theory the time coordinate $x^0$ is equal to $\tau$ in all frames - that's what makes it Newtonian instead of relativistic mechanics.

 

[DrGreg]

It's surprising see how it looks the same as the incorrect one. It seems that the only difference is that time is accounted as one of the directions, i.e. the force becomes a Minkowski-four-vector. Did I get this right?

There are two other differences: coordinate time $t$ has been replaced by proper time $\tau$ and the coordinate derivative $\rm{d}$ has been replaced by the absolute or invariant derivative (a.k.a. covariant derivative) $\rm{D}$.

Why can't one use this lorentz invariant form in Newton's theory?

In Newton's theory there is only one time (so $t = \tau$), and in an inertial frame $\rm{d}$ and $\rm{D}$ are the same.

 

[davidge]

If you do, you don't have Newton's theory. In Newton's theory the time coordinate $x^0$ is equal to $\tau$ in all frames - that's what makes it Newtonian instead of relativistic mechanics.

I see. But let's say the clock in one frame is off by say, 5, with respect to the clock in the other frame. So $t' = t+5$. Still this is non relativistic, correct?

 

[DrGreg]

I see. But let's say the clock in one frame is off by say, 5, with respect to the clock in the other frame. So $t' = t+5$. Still this is non relativistic, correct?

Yes. To be more precise, I should have said that in Newtonian theory clocks can only differ by a constant offset, or, to put it another way, there's only one ${\rm d}t$.

 

[davidge]

So to summarize what I learned on this thread,

- A reference frame being inertial means only that the law of inertia holds as pointed out by @PeterDonis
- Newton's equation of motion is not invariant under Lorentz transformations, which means that Newton's second law is not invariant under Lorentz transformations accounting for what @Nugatory said
- To get the correct equation of motion one needs to come up with the equation @DrGreg presented, which is no longer the Newton's equation of motion

But this brings another question: why does the laws of nature which I suppose, should include Newton's second law, are the same in inertial frames if the equation that describes this law, i.e. the corresponding Newton's second law equation doesn't remain invariant?

 

[DrGreg]

But this brings another question: why does the laws of nature which I suppose, should include Newton's theory, are the same in inertial frames if the equation that describes this theory, i.e. the corresponding Newton's equation of motion doesn't remain invariant?

Saying that "the laws of nature are the same in all inertial frames" doesn't specify what the "laws of nature" actually are. Newton's 2nd Law as usually expressed is, for low speeds, approximately but not exactly the same as the relativisitically correct law, so it is just a case of reformulating the laws so that they are approximately the same as Newton's at low speeds and low tidal gravity, but which also satisfy the postulates of relativity.

 

[davidge]

Ah, ok. So this would be a case that Newton's statement remains correct, but his mathematical description not? I mean, "force causes the velocity of a object to change" remains invariant, but the corresponding second equation must be adjusted?