# Why does Newtonian dynamics break down at the speed of light

There's no stuff gets heavier at high speeds term in Newtons equations.

Compared to F=dp/dt
Exacty - Newton's second law: "The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed" together with the definition: "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly". Thus, in equation form : dp/dt = F

What turned out to be inexact in Newton's theory, at high speeds, were his assumptions about mass, time and length.

What turned out to be inexact in Newton's theory, at high speeds, were his assumptions about mass, time and length.
Which assumptions about mass do mean?

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It's true that dp/dt = F relativistically, where bold indicates 3-vectors. This can be derived from the more general dpμ/dτ=Fμ, which gets us into the 4-force that Chet was talking about. However, none of these terms is what it is in Newtonian mechanics.

Symon has a nice treatment of this in Chapter 14.

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It's true that dp/dt = F relativistically, where bold indicates 3-vectors. This can be derived from the more general dpμ/dτ=Fμ, which gets us into the 4-force that Chet was talking about. However, none of these terms is what it is in Newtonian mechanics.

Symon has a nice treatment of this in Chapter 14.
Yes. For me, this was all I needed to be satisfied that the relationship was recovered intact in SR (even if the terms in component form were not the same). Seeing it satisfied in vector form made me very happy.

Chet

Yes. For me, this was all I needed to be satisfied that the relationship was recovered intact in SR (even if the terms in component form were not the same). Seeing it satisfied in vector form made me very happy.

Chet
I understand Chet's point now. However I am still under the impression that this formal similarity between the equations for force has more to do with the way we define forces, i.e.with the way we represent interactions in the mathematical formalism of physical theories. This is regarding the right side of equations of the type "F= (some def. of force)".
I am not sure why the form of the force law for a charged particle interacting with an electromagnetic field is so formally similar though, I should take another look at Chet's reference.

Personally I prefer using the variational approach when it comes to relativistic mechanics.

Which assumptions about mass do mean?
"DEFINITION I.
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter."

With SR this simple and straightforward definition of mass had to be abandoned. According to SR, hot water weighs more than cold water. On top of that, a water molecule does not even weigh the same as the sum of its atoms.

It's true that dp/dt = F relativistically, where bold indicates 3-vectors. This can be derived from the more general dpμ/dτ=Fμ, which gets us into the 4-force that Chet was talking about. However, none of these terms is what it is in Newtonian mechanics.
That depends on what you mean with "these terms". F=dp/dt applies both for classical mechanics and relativity but the the different transformations result in different expressions for F.

"DEFINITION I.
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
[...]"

With SR this simple and straightforward definition of mass had to be abandoned.
I don't see why. It is not very helpful and today we rather use it in reverse to define density (as mass per volume) but that does not mean that it is wrong.

According to SR, hot water weighs more than cold water.
If the volume remains constant then heating the water will increase its density and its mass by the same factor. If you keep the density constant than the volume will be increase in the same way as mass. If nothing remains constant then the situation gets complicate but it will be always full consistent with definition 1.

On top of that, a water molecule does not even weigh the same as the sum of its atoms.
How does this collide with Newton's concept of mass?

[..] If the volume remains constant then heating the water will increase its density and its mass by the same factor. If you keep the density constant than the volume will be increase in the same way as mass. If nothing remains constant then the situation gets complicate but it will be always full consistent with definition 1.
I don't follow you. According to SR, a constant number of water molecules (amount of matter) will increase in mass when heated due to increased kinetic energy. According to Newton's mechanics the mass is fixed. Of course, a discussion of m=E/c2 belongs in the relativity forum.
How does this [a water molecule does not even weigh the same as the sum of its atoms] collide with Newton's concept of mass?
According to Newton's mechanics, the mass of all particles together ("condensed" or other) equals the sum of all particles separately. The fact that this is not exactly the case is therefore called mass "defect".

According to SR, a constant number of water molecules (amount of matter) will increase in mass when heated due to increased kinetic energy. According to Newton's mechanics the mass is fixed.
That does not result from definition 1.

According to Newton's mechanics, the mass of all particles together ("condensed" or other) equals the sum of all particles separately.
By replacement of Galilean transformation by Lorentz transformation Newton's quantity of matter turns into relativistic mass and relativistic mass is additive.

The fact that this is not exactly the case is therefore called mass "defect".
That's another topic. Mass defect is the difference of the total rest mass of a system (including binding energy and kinetic energies) and the sum of the rest masses of its sub systems (excluding binding energy between them). That doesn't contradict Newton because he didn't make corresponding claims.

There's no "rest mass" becouse there's no "relaivistic mass". Mass doesn't depend on velocity. Therefore, there is just mass .

• brainpushups and vanhees71
That does not result from definition 1.
It is stated that mass equals "amount of matter" and it is implied that it is not a function of temperature or speed. You will search in vain for any such a relationship in classical mechanics.
By replacement of Galilean transformation by Lorentz transformation Newton's quantity of matter turns into relativistic mass and relativistic mass is additive. [..]
We disagree about how to present the same facts; I'm afraid that we will have to agree to disagree. Newton's quantity of matter is not a function of speed. The Newtonian definitions and laws resulted in (or gave an explanation for) the "Galilean transformations" which Newton's mechanics assumed to be correct. The "relativistic mass" and "invariant mass" concepts came about because Newton's mass concept - as well as his time and length concepts - could not be maintained in relativity theory.

In order to stay within the bounds of this forum, I won't comment on claims about relativity theory in this thread. However, if you can show mass increase due to temperature increase in Newton's mechanics, I'll be happily corrected!

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Even Newton's (well, Newton's and Euler's) definitions of mass are a little mixed up. Indeed, Newton defines mass as the amount of matter, but the operational definition of mass (in terms of the second law) is that mass is a measure of inertia.

Also the definition of mass is often used differently in relativity. Most authors now prefer to define the mass of an object the invariant mass while others use variable mass.

It is stated that mass equals "amount of matter" and it is implied that it is not a function of temperature or speed.
Quantity of matter is Newton's name for his concept of mass but not its definition. You must not confuse it with amount of substance or similar modern concepts. I totally agree with brainpushup that quantity of matter is a measure for inertia.

Newton's quantity of matter is not a function of speed.
Not in classical mechanics, but in special relativity.

The Newtonian definitions and laws resulted in (or gave an explanation for) the "Galilean transformations"
That does not apply to definition 1-2 and lex 1-3. They work with Lorentz transformation as well.

The "relativistic mass" and "invariant mass" concepts came about because Newton's mass concept - as well as his time and length concepts - could not be maintained in relativity theory.
In special relativity the relativistic mass directly results from Newton's quantity of matter (as defined by definition 2, lex 2 and lex 3).

In order to stay within the bounds of this forum, I won't comment on claims about relativity theory in this thread.
Than you must not post in this thread at all. The break down of Newtonian dynamics at the speed of light is well outside classical mechanics.

However, if you can show mass increase due to temperature increase in Newton's mechanics, I'll be happily corrected!
Why should I do that? It is sufficient that Newton's mechanics does not exclude an increase of quantity of matter due to temperature increase. In addition it is in agreement with historical concepts of heat (e.g. phlogiston).

Also the definition of mass is often used differently in relativity. Most authors now prefer to define the mass of an object the invariant mass while others use variable mass.
That's why it is better to use Newton's term quantity of matter to avoid confusions with mass (zoki85 already got caught in that trap).