Why does Newtonian dynamics break down at the speed of light

[nisarg]

I tried searching the web for this topic but got an answer like "formulae used in classical mechanics are approximations or simplifications of more accurate formulae such as the ones in quantum mechanics and special relativity". My question is that why do the laws of Sir Isaac Newton no longer apply to objects at the speed of light? Is it the formulae that are causing the problem or the laws?

I really need a detailed explanation to understand this topic thoroughly, so if someone could help me on this, I would be more than grateful.

Thanks

 

[Borg]

Take a look at the Wikipedia article on General Relativity to start. There is a section that explains going from classical Newtonian mechanics to General Relativity.

 

[Vanadium 50]

. My question is that why do the laws of Sir Isaac Newton no longer apply to objects at the speed of light? Is it the formulae that are causing the problem or the laws?

The laws are low velocity approximations. At higher and higher velocities, the laws get worse and worse.

 

[DrStupid]

My question is that why do the laws of Sir Isaac Newton no longer apply to objects at the speed of light?

Do they? Galilean transformation fails at the speed of light, but Newton's laws of motion (in their original form) still apply. If it makes sense to use forces for photons is another question.

 

[mfb]

My question is that why do the laws of Sir Isaac Newton no longer apply to objects at the speed of light? Is it the formulae that are causing the problem or the laws?

We just happen to live in a universe where Newtonian physics is not exact. It is perfectly possible to imagine a world where the laws are exact at all speeds (particle physics and some other fields would get problems , but let's ignore the microscopic part here), but experiments show we do not live in such a world.

but Newton's laws of motion (in their original form) still apply.

Acceleration is not parallel to force in general. How does that agree with Newtonian physics?

 

[DrStupid]

Acceleration is not parallel to force in general. How does that agree with Newtonian physics?

With replacement of Galilei transformation by Lorentz transformation Newton's "quantity of matter" becomes velocity dependent. In the result acceleration is no longer parallel to force.

 

[Chestermiller]

I'm not too experienced with relativity so I'm not sure, but isn't the 4-force parallel to the 4-acceleration (unless the rest mass is changing)?

Chet

 

[mfb]

With replacement of Galilei transformation by Lorentz transformation Newton's "quantity of matter" becomes velocity dependent. In the result acceleration is no longer parallel to force.

A velocity-dependent scalar mass is not sufficient, you would need some sort of "vector mass". And I think that is beyond Newton's equation of motion. Even the Lorentz transformations on their own are beyond Newton's physics.

 

[DrStupid]

A velocity-dependent scalar mass is not sufficient, you would need some sort of "vector mass".

The velocity-dependent scalar mass

$m = \frac{{m_0 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}$

results in

$a = \left( {\frac{F}{{m_0 }} - v \cdot \frac{{v \cdot F}}{{m_0 \cdot c^2 }}} \right) \cdot \sqrt {1 - \frac{{v^2 }}{{c^2 }}}$

There is no need for some sort of "vector mass".

Even the Lorentz transformations on their own are beyond Newton's physics.

Of course it is. That's why I limited my statement to Newton's laws of motion.

 

[mfb]

$a = \left( {\frac{F}{{m_0 }} - v \cdot \frac{{v \cdot F}}{{m_0 \cdot c^2 }}} \right) \cdot \sqrt {1 - \frac{{v^2 }}{{c^2 }}}$

Okay, if you add those extra terms - I would not call this "Newton's laws of motion" any more.

 

[DrStupid]

Okay, if you add those extra terms - I would not call this "Newton's laws of motion" any more.

There are no extra terms.

 

[dextercioby]

The 2nd law fails simply because it allows accelerating moving objects to increase their speed indefinitely, but as per the current valid theories and experimental results, nothing in our universe can move at a speed faster than c, when its speed is measured in an inertial ref. frame.

 

[mfb]

There are no extra terms.

Compared to a=F/m?

 

[DrStupid]

Compared to a=F/m?

Compared to F=dp/dt

 

[Vanadium 50]

There is no need for some sort of "vector mass".

You've rolled it into the second term in the pre-factor.

 

[DrStupid]

You've rolled it into the second term in the pre-factor.

Which second term of which pre-factor?

 

[Vanadium 50]

The part with the dot product. That makes it directional.

 

[DrStupid]

The part with the dot product. That makes it directional.

Are you confusing the equation for acceleration with the equation for quantity of matter (we better do not use the term mass at this place)? The latter does not contain such a part.

 

[Chestermiller]

As I said in post number 7, if expressed in terms of the 4-force and 4-acceleration, Newton's second law is recovered intact.

Chet

 

[brainpushups]

My question is that why do the laws of Sir Isaac Newton no longer apply to objects at the speed of light? Is it the formulae that are causing the problem or the laws?

Newton made a few assumptions about nature that turned out to be incorrect. For example, Newton's conception of time in the definitions given in the Principia as a quantity that moves forward independently without regard to motion (I'm paraphrasing here) was questioned later by Mach who influenced Einstein. Einstein also knew that Maxwell's equations predicted electromagnetic waves that all traveled with the same speed... but relative to what? After the rejection of the lumineferous ether largely due to the Michelson-Morley experiment Einstein proposed the two postulates of special relativity - one of which is that the speed of light is the same for all observers regardless of their state of motion. One of the consequences of this postulate (which does not suppose that time runs the same for everyone) is that the amount of time elapsed depends on an observers state of motion. This (and other consequences) of special relativity are only important when the speeds of objects approach the speed of light. If the speeds are low then the predictions made by SR reduce to Newtonian mechanics.

So... to answer the question. It is the axiomatic assumptions that are 'causing the problems'

As others have said, Newton's laws are still applicable in SR if you change the definition of force and momentum to be their four-vector definitions. However, in my limited experience with SR I've noticed that the concept of force (in the Newtonian sense) is not very convenient simply because of how messy this would get when applying the Lorentz transformations. The form of the laws look the same when using four-vectors (which is probably one of the reasons that four-momentum was defined the way it was!), but I would argue that this isn't really Newton's laws anymore.

 

[Chestermiller]

As others have said, Newton's laws are still applicable in SR if you change the definition of force and momentum to be their four-vector definitions. However, in my limited experience with SR I've noticed that the concept of force (in the Newtonian sense) is not very convenient simply because of how messy this would get when applying the Lorentz transformations. The form of the laws look the same when using four-vectors (which is probably one of the reasons that four-momentum was defined the way it was!), but I would argue that this isn't really Newton's laws anymore.

I used to feel the same way, thinking that the 4-force was just a contrived entity, designed specifically to recover Newton's 2nd law within the framework of SR. However, that feeling was dispelled when I saw the dazzling development in MTW in which they derived the equation for the 4-force acting on a stationary or moving charge within an electric and/or magnetic field and showing that it was described, independent of ma, by the magnitude of the charge times the contraction of the Faraday tensor with the 4-velocity vector. Did you not have the same response when you studied this?

Chet

 

[brainpushups]

Cool. I have not seen that derivation. I'll have to check it out.

 

[Cruz Martinez]

I used to feel the same way, thinking that the 4-force was just a contrived entity, designed specifically to recover Newton's 2nd law within the framework of SR. However, that feeling was dispelled when I saw the dazzling development in MTW in which they derived the equation for the 4-force acting on a stationary or moving charge within an electric and/or magnetic field and showing that it was described, independent of ma, by the magnitude of the charge times the contraction of the Faraday tensor with the 4-velocity vector. Did you not have the same response when you studied this?

Chet

I felt this way for a short time too. For me the four-force concept became clear when I started thinking about it as just a measure of the deviation of a particle's path from geodesic motion. Forces are defined that way too in Newtonian mechanics after all, as a measure of deviation from inertial motion.

 

[Chestermiller]

I felt this way for a short time too. For me the four-force concept became clear when I started thinking about it as just a measure of the deviation of a particle's path from geodesic motion. Forces are defined that way too in Newtonian mechanics after all, as a measure of deviation from inertial motion.

That would apply more to acceleration than to force.

Chet

 

[Cruz Martinez]

That would apply more to acceleration than to force.

Chet

yes, you're right, I will have to elaborate more on that thought.

If there's acceleration then we have force, but that alone can't account for the whole thing since different particles deviate from inertial motion differently when they interact with the same fields.
But still forces are a measure of how particles deviate from inertial motion,
I guess it is safe to say that force is a measure of how much a particle's motion is not inertial, once you take into account the dynamical parameters of the particle, what do you think?

 

[Puma]

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

 

[harrylin]

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.

 

[DrStupid]

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?

 

[Vanadium 50]

It's true that dp/dt = F relativistically, where bold indicates 3-vectors. This can be derived from the more general dp_μ/dτ=F_μ, 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.

 

[Chestermiller]

It's true that dp/dt = F relativistically, where bold indicates 3-vectors. This can be derived from the more general dp_μ/dτ=F_μ, 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

 

[Cruz Martinez]

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.

 

[HomogenousCow]

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

 

[harrylin]

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.

 

[DrStupid]

It's true that dp/dt = F relativistically, where bold indicates 3-vectors. This can be derived from the more general dp_μ/dτ=F_μ, 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?

 

[harrylin]

[..] 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".