B Newtonian vs Relativistic Mechanics

[Grimble]

It is said many times, from the days of Einstein, Minkowski and Poincaré, that Classic or Newtonian Mechanics are not consistent with motion at relativistic speeds, that a new relativistic mechanics is needed, viz.
Albert Einstein: … the apparent incompatibility of the law of propagation of light with the principle of relativity […] has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows: ⁠1

1. The time-interval (time) between two events is independent of the condition of motion of the body of reference.
2. The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.

Hermann Minkowski: I would like to show you at first, how we can arrive – from mechanics as currently accepted – at the changed concepts about time and space, by purely mathematical considerations. […]However, it is to be remembered that a modified mechanics will hold now…⁠2

Henri Poincaré: From all these results, if they were to be confirmed, would issue a wholly new mechanics which would be characterized above all by this fact, that there could be no velocity greater than that of light, any more than a temperature below that of absolute zero. For an observer, participating himself in a motion of translation of which he has no suspicion, no apparent velocity could surpass that of light, and this would be a contradiction, unless one recalls the fact that this observer does not use the same sort of timepiece as that used by a stationary observer, but rather a watch giving the “local time.[..] Perhaps, too, we shall have to construct an entirely new mechanics that we only succeed in catching a glimpse of, where, inertia increasing with the velocity, the velocity of light would become an impassable limit.⁠3

But just what are the differences? Is there a description?1 Albert Einstein (1879–1955). Relativity: The Special and General Theory. 1920.
XI The Lorentz Transformation.

2 Raum und Zeit(1909), Jahresberichte der Deutschen Mathematiker-Vereinigung, 1-14, B.G. Teubner
A Lecture delivered before the Naturforscher Versammlung (Congress of Natural Philosophers) at Cologne — (21st September, 1908).

3 Poincaré, Henri (1904/6), "The Principles of Mathematical Physics", Congress of arts and science, universal exposition, St. Louis, 1904 1, Boston and New York: Houghton, Mifflin and Company, pp. 604–622

 

[vanhees71]

Sure there is a description. Have a look at my relativity FAQ

http://fias.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[Megaquark]

The differences between Newtonian and Relativistic physics? The main one is that in Newton's physics, cause and effect are instantaneous. In Relativity, cause and effect take time...in relativity there is no such thing as instantaneous action at a distance. There are all sorts of implications for having a finite speed of causality, and Einstein explores these consequences in his special theory. The main ones are that different observers don't agree upon the time separation, the space separation, or the simultaneity of events.

 

[Dale]

But just what are the differences? Is there a description?

The best formulation of the differences is in terms of four vectors. E.g. Newton's 2nd law $f=dp/dt$ where f and p are the force and momentum three vectors and t is Galilean time changes to $F=dP/d\tau$ where F and P are the four-force and four-momentum four-vectors and $\tau$ is the proper time.

 

[vanhees71]

One should add that there are constraints in the relativistic case since for a classical point particle the mass-shell condition
$$P_{\mu} P^{\mu}=m^2 c^2=\text{const}$$
should be fulfilled. Taking the derivative wrt. $\tau$ gives
$$\dot{P}_{\mu} P^{\mu}=0$$
and thus the four-force must fulfill
$$F_{\mu} p^{\mu}=0.$$
So together there are only $3$ independent coordinates, not $4$, as in non-relativistic mechanics.

 

[DrStupid]

But just what are the differences? Is there a description?

Newton: Galilean transformation
Einstein: Lorentz transformation

Everything else in SR results from this difference.

 

[Grimble]

Allow me to ask a question here...
When I created this thread, the system insisted I insert a prefix at the start for the level. I put a B as when you start talking of:

the four-force and four-momentum four-vectors and ττ\tau is the proper time.

classical point particle the mass-shell condition

then I am lost - I had a 'high school education' I guess you would call it - I am from the UK.

I am thinking of this in a very basic way currently. I understand Newtonian Mechanics - that is what we were taught at school - very simple and straight forward. What I am looking for now, is some help in finding out which of Newton's laws are changed by Relativity.
Newtons Mechanics are represented in Cartesian coordinates with a constant time factor that is the same throughout.
In Relativistic Mechanics, I understand we are considering Time as the fourth dimension, rather than as a common standard, but how is this depicted in diagrams? In the Minkowski diagrams it seems to be the rotation of the moving frame's time access? OK but what of the other three axes? Are they not still orthogonal?

 

[DrStupid]

What I am looking for now, is some help in finding out which of Newton's laws are changed by Relativity.

His law of gravitation has been invalidated. It is inconsistent with Lorentz transformation and can't be fixed to fit into relativity. That's why Einstein developed general relativity.

 

[Ibix]

I am thinking of this in a very basic way currently. I understand Newtonian Mechanics - that is what we were taught at school - very simple and straight forward. What I am looking for now, is some help in finding out which of Newton's laws are changed by Relativity.

A lot of concepts persist, but are always modified. For example, momentum is conserved just as in Newtonian physics, but the expression for momentum is different. Forces get quite complicated, enough so that most people don't use them. As DrStupid says, Newtonian gravity is completely incompatible with relativity because the propagation speed of Newtonian gravity is infinite, and relativity requires that nothing travel faster than light.

Newtons Mechanics are represented in Cartesian coordinates with a constant time factor that is the same throughout.
In Relativistic Mechanics, I understand we are considering Time as the fourth dimension, rather than as a common standard, but how is this depicted in diagrams? In the Minkowski diagrams it seems to be the rotation of the moving frame's time access? OK but what of the other three axes? Are they not still orthogonal?

The really tricky concept (well, one of them) to get your head round in relativity is that coordinates don't really mean anything. They're just a systematic labelling system for points in spacetime. Einstein used a physical metaphor to create his coordinate system - a 3d grid of rigid rods with clocks at every intersection - and this does indeed define a system of coordinates whose axes (all four of them, although Minkowski was the first to realize this) are mutually orthogonal. However, the concept of orthogonality has to be slightly more general than the one from Euclidean geometry, because the rules of Euclidean geometry do not apply to spacetime. So when you see a Minkowski diagram that shows the time and space axes from a moving frame "scissored together", they are still orthogonal. It's just not possible to draw them orthogonal on a Euclidean plane for the same reason you can't draw an accurate flat map of the whole world

I'm sorry if that is more confusing than helpful. Did you come across matrices and vectors at school? (I'm also UK based, but I've no idea when you went through the school system or how the curriculum has varied over time.)

 

[PeroK]

Allow me to ask a question here...
When I created this thread, the system insisted I insert a prefix at the start for the level. I put a B as when you start talking of:then I am lost - I had a 'high school education' I guess you would call it - I am from the UK.

I am thinking of this in a very basic way currently. I understand Newtonian Mechanics - that is what we were taught at school - very simple and straight forward. What I am looking for now, is some help in finding out which of Newton's laws are changed by Relativity.
Newtons Mechanics are represented in Cartesian coordinates with a constant time factor that is the same throughout.
In Relativistic Mechanics, I understand we are considering Time as the fourth dimension, rather than as a common standard, but how is this depicted in diagrams? In the Minkowski diagrams it seems to be the rotation of the moving frame's time access? OK but what of the other three axes? Are they not still orthogonal?

Have you tried to learn some Special Relativity? It doesn't require much maths but obviously has some conceptual hurdles to overcome.

I'm not so sure that a list of ways that relativity is different from classical mechanics is that helpful. Relativity contains classical mechanics as a special case where speeds are small compared to the speed of light. So, in a sense, everything that holds for classical mechanics holds in SR, but only for low speeds.

A good example of this is the relativistic kinetic energy of a particle, which is:

$KE = (\gamma - 1) mc^2$ where $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$ and $v$ is the speed of the particle.

Now, using the binomial theorem:

$\gamma = (1 - v^2/c^2)^{-1/2} = 1 - \frac{1}{2}(-v^2/c^2) + \dots$

Hence:

$KE = (\gamma - 1) mc^2 = (\frac{1}{2}(v^2/c^2) + \dots)mc^2 = \frac{1}{2}mv^2 + \dots$

Where all other terms are small if $v$ is much less than $c$.

So, the relativistic kinetic energy $(\gamma - 1) mc^2$ actually reduces to the familiar $\frac{1}{2}mv^2$ in the special case of speeds much less than $c$.

I'd encourage you to be more interested in why relativistic KE is this seemingly unexpected expression, rather than worry about how it is different from the classical expression for KE.

 

[Dale]

When I created this thread, the system insisted I insert a prefix at the start for the level. I put a B as when you start talking of:

My apologies. I did not properly consider the requested "B" when writing my response.

I guess that I should have simply said that yes there are differences and yes the differences are completely well described mathematically. The core difference is the Galilean transform vs the Lorentz transform, and there are many elegant and powerful mathematical tools for expressing that difference.

I understand we are considering Time as the fourth dimension, rather than as a common standard, but how is this depicted in diagrams? In the Minkowski diagrams it seems to be the rotation of the moving frame's time access? OK but what of the other three axes? Are they not still orthogonal?

Yes, but it is hard to draw four perpendicular axes on a piece of paper. So usually we drop two of the spatial axes (usually y and z)

 

[ogg]

Just to clear up one point: Special Relativity doesn't deal with gravity directly, it presumes a Newtonian gravitational force (or perhaps it would be better to say it presumes *nothing* about gravity, and any assumptions you use are external to SR). This thread is the first I've read about modified Newtonian force being incompatible with "relativity". I don't believe it is, but I'm not highly proficient in that area. Specifically, I believe that IF you modify force to propagate at c (or less), then you can use Newtonian Physics (in low velocity, low gravity contexts). I've no doubt that this neo-Newtonian approach is quite incompatible with GR, but I do question its inapplicability to SR. Anyway, the basic difference between SR and Classical physics is that in SR, each observer (each particle) has its own clock (proper time) which does *not* match up with any other clock (unless the two are in inertial frames, differences in speed (assuming they're constant) can easily be incorporated; but NOT accelerations (changes in direction or velocity)). In fact, we now have clocks which can measure differences in height of about 1 foot (on Earth's surface); your feet are experiencing a slightly different time than your head. Without college physics, perhaps the two most useful formulas of SR to play with are Mass = M(0)÷√(1-(v/c)²) where M(0) is the rest mass (the mass measured in an inertial frame with zero velocity) v is the velocity of the observer and c is the speed of light. At v/c=0.9 we have M/M(0) ≈ 2.3 or an object traveling at 0.9c is more than twice as heavy as the same object at rest. The other concerns time: t = t(0)*√(1-(v/c)²). Meaning that a one second tick of a clock traveling at 0.9c would seem to an observer at rest to take 2.3 seconds (or in an observer's 1 second of elapsed time, the moving object would experience 0.44 seconds). We could also write an equation for the length of objects traveling very fast, since space and time are "mixed" in relativity, a distortion of time implies that distances (lengths) will also be changed.

 

[PeroK]

Just to clear up one point: Special Relativity doesn't deal with gravity directly, it presumes a Newtonian gravitational force (or perhaps it would be better to say it presumes *nothing* about gravity, and any assumptions you use are external to SR). This thread is the first I've read about modified Newtonian force being incompatible with "relativity". I don't believe it is, but I'm not highly proficient in that area. Specifically, I believe that IF you modify force to propagate at c (or less), then you can use Newtonian Physics (in low velocity, low gravity contexts). I've no doubt that this neo-Newtonian approach is quite incompatible with GR, but I do question its inapplicability to SR. Anyway, the basic difference between SR and Classical physics is that in SR, each observer (each particle) has its own clock (proper time) which does *not* match up with any other clock (unless the two are in inertial frames, differences in speed (assuming they're constant) can easily be incorporated; but NOT accelerations (changes in direction or velocity)). In fact, we now have clocks which can measure differences in height of about 1 foot (on Earth's surface); your feet are experiencing a slightly different time than your head. Without college physics, perhaps the two most useful formulas of SR to play with are Mass = M(0)÷√(1-(v/c)²) where M(0) is the rest mass (the mass measured in an inertial frame with zero velocity) v is the velocity of the observer and c is the speed of light. At v/c=0.9 we have M/M(0) ≈ 2.3 or an object traveling at 0.9c is more than twice as heavy as the same object at rest. The other concerns time: t = t(0)*√(1-(v/c)²). Meaning that a one second tick of a clock traveling at 0.9c would seem to an observer at rest to take 2.3 seconds (or in an observer's 1 second of elapsed time, the moving object would experience 0.44 seconds). We could also write an equation for the length of objects traveling very fast, since space and time are "mixed" in relativity, a distortion of time implies that distances (lengths) will also be changed.

That's a helluva way to clear up one point. You even sneaked relativistic mass in there while no one was looking! And, you give the impression of motion and rest being absolute: "a clock traveling at 0.9c", "an observer at rest".

 

[Ibix]

Just to clear up one point: Special Relativity doesn't deal with gravity directly, it presumes a Newtonian gravitational force (or perhaps it would be better to say it presumes *nothing* about gravity, and any assumptions you use are external to SR). This thread is the first I've read about modified Newtonian force being incompatible with "relativity". I don't believe it is, but I'm not highly proficient in that area.

Special relativity assumes no gravity at all. Strictly, it's only valid in situations where gravity is negligible.

There were attempts to create modified Newtonian gravity theories that were compatible with special relativity, but they fell by the wayside when general relativity came along and predicted things like the precession of Mercury accurately. I don't know much about them. If you are moving slowly (<<c) and don't get too close to a black hole, Newton will do you fine (I am told NASA use nothing else, and pull off feats comparable to hitting a dust grain in Paris with another thrown from London). Anywhere he doesn't work you should probably go for a full general relativistic treatment, on the basis that it is well understood and you know that you're using the best tools available.

Without college physics, perhaps the two most useful formulas of SR to play with are Mass = M(0)÷√(1-(v/c)²) where M(0) is the rest mass (the mass measured in an inertial frame with zero velocity) v is the velocity of the observer and c is the speed of light.

Please don't. Most authors since about the 1950s use "mass" to mean rest mass and don't mention the relativistic mass. "Relativistic mass" is the same thing as "total energy", and it's confusing to use the term. Not to mention that so many people read this formula and immediately ask "so can I turn into a black hole if I move fast enough?". There's an FAQ for that...

We could also write an equation for the length of objects traveling very fast, since space and time are "mixed" in relativity, a distortion of time implies that distances (lengths) will also be changed.

There is no distortion involved in special relativity. All the "frames of reference" stuff is is a change of coordinates, closely analogous to turning a map so that the streets match the orientation you are physically using.

 

[Dale]

This thread is the first I've read about modified Newtonian force being incompatible with "relativity". I don't believe it is,

Consider a 1 kg object traveling at 1 m/s less than c subject to a force of 1 N for a duration of 2 s.

 

[pervect]

Just to clear up one point: Special Relativity doesn't deal with gravity directly, it presumes a Newtonian gravitational force (or perhaps it would be better to say it presumes *nothing* about gravity, and any assumptions you use are external to SR). This thread is the first I've read about modified Newtonian force being incompatible with "relativity". I don't believe it is, but I'm not highly proficient in that area. Specifically, I believe that IF you modify force to propagate at c (or less), then you can use Newtonian Physics (in low velocity, low gravity contexts).

You might want to read a bit of history then. A good place to start would be https://en.wikipedia.org/w/index.php?title=Scalar_theories_of_gravitation&oldid=710756612

(The version is included so we can all talk about the same version of the wiki article).

The good news is that indeed one can come up with a consistent relativistic theory of gravity that is similar to Newton's theory in the sense that it is a scalar theory of gravity. Whether or not that's the route you had in mind is an open question, but unless you're more specific about this nebulous theory that you say you have in mind, it doesn't seem to be too productive to go down a lot of different paths to guess what it might be. So let's stick to scalar theories of gravity for the time being as a good example that illustrates the history of the developments that eventually led to GR. If you have some other approach in mind, a bit of reading is likely to find that someone has tried it.

Scalar theories of gravity are attractive and familiar because the gravitational field is represented by a scalar potential (i.e. a rank 0 tensor), rather than the high rank tensors that GR uses. The bad news is that the base theory that arises from this scalar field approach, Nordstrom's theory, isn't consistent with observation. For instance, it predicts no deflection of light by gravity. I'm not aware offhand of what other possible variants of scalar theories of gravity might exist that try to "fix" this problem with Nordstrom's theory, but I do know that none of them panned out. Einstein eventually developed General Relativity, which is not a scalar theory at all, and to date experiment has borne him out.

 

[PeterDonis]

it presumes a Newtonian gravitational force

No, it doesn't. It presumes flat spacetime, which is incompatible with gravity being present at all. In other words, SR is only valid in scenarios where gravity is negligible.

I believe that IF you modify force to propagate at c (or less), then you can use Newtonian Physics (in low velocity, low gravity contexts).

No, you can't. Newtonian gravity with a finite propagation speed of $c$ is grossly inconsistent with observation: for example, there are no closed orbits or even almost-closed orbits with small precession like the orbits in GR; and the Newtonian force points in the wrong direction, i.e., it exhibits aberration far in excess of what is observed. I suggest reading Carlip's classic paper on aberration and the speed of gravity, which discusses these aspects of "modified Newtonian gravity" as a preparation for showing how GR in the Newtonian limit actually works:

http://arxiv.org/abs/gr-qc/9909087

 

[Grimble]

Yes, but it is hard to draw four perpendicular axes on a piece of paper. So usually we drop two of the spatial axes (usually y and z)

Yes that is a good way to keep it simple, but it does raise a question for me; I understand how the time axis for the moving frame of reference is rotated, relativistically rather than Classically, but does that mean the other axes of the moving frame are rotated too to maintain orthogonality (is that even a word? hehehe), for if they were rotated too then the shared x-axis would no longer apply...

Please help me for I am trying to understand how this works, for the one real change in the postulates for SR over classical mechanics is the limit of 'c'. And the differences that then appear in time and length as a consequence.

Is it true that the whole of relativity is founded upon those two simple, yet fundamental postulates: the first reassuring us that we are dealing with a consistent framework that is the same everywhere - homogeneous and isometric - are I think how that is labelled; while the second is the the very innocent sounding light moves at a constant rate.
The invariance of the speed of light means that Newtonian diagrams of mechanics won't work at near light speeds as standard vector addition may result in speeds greater than 'c'.

So I was just wondering what those updates or changes to Newtonian mechanics were and how they are shown in diagrams.

Take, for instance, our old friend the moving light clock from a Newtonian viewpoint. If the light has traveled 1 ls vertically in the clock while the clock is moving at 0.6c away from the observer, by the time the light in the clock reaches the mirror 1 ls away for an observer moving with the clock, for the static observer that light would have traveled I ls vertically and 0.6 ls horizontally - or 1.25 ls diagonally - still in the time of i second. (standard time dilation diagram)
Now if that is drawn from a relativistic perspective what changes are made to axes or coordinate choices? How is the diagram altered to cope with the changed perspective.

(I'm sorry I can't see how to add a digram... is there an easy way?)

 

[PeroK]

So I was just wondering what those updates or changes to Newtonian mechanics were and how they are shown in diagrams.

As I said above, I think an attempt to understand SR as a set of updates to Newtonian Mechanics is doomed to failure. You need to tackle SR itself directly. Once you have mastered SR, you can sit back and compare it with Newton to your heart's content.

 

[Dale]

Yes that is a good way to keep it simple, but it does raise a question for me; I understand how the time axis for the moving frame of reference is rotated, relativistically rather than Classically, but does that mean the other axes of the moving frame are rotated too to maintain orthogonality (is that even a word? hehehe), for if they were rotated too then the shared x-axis would no longer apply

So look at the Lorentz transform equation. What does it tell you about the rotation of the y and z axes?

EQ 1346-1349
http://farside.ph.utexas.edu/teaching/em/lectures/node109.html

So I was just wondering what those updates or changes to Newtonian mechanics were and how they are shown in diagrams.

The same link that I posted above also contains the Galilean transform. It is a very useful exercise to plot both transforms for, say v=0.6 c. If you have both plotted then it becomes graphically easy to see the differences.

 

[DrStupid]

This thread is the first I've read about modified Newtonian force being incompatible with "relativity".

Newtonian force (F=dp/dt) doesn't need to be modified to be compatible with relativity. It is Newton's law of gravity which is incompatible with relativity.

 

[haushofer]

I'd say that in both relativity-theories inertial observers measure the same speed of light. However, in Newtonian/Galilean relativity this speed is infinite (hence the appearance of an absolute time) and in Einstein's theory this speed is finite. I think that's the crucial difference.

 

[Ibix]

I'd say that in both relativity-theories inertial observers measure the same speed of light. However, in Newtonian/Galilean relativity this speed is infinite (hence the appearance of an absolute time) and in Einstein's theory this speed is finite. I think that's the crucial difference.

Something that probably needs explaining here for the OP. Both Einstein and Newton envisage an "invariant speed", a speed that is exactly the same to all observers. In Newtonian relativity, that speed is infinity - if something passes me at infinite speed, it doesn't matter whether I consider myself to be stationary, moving at 60mph, or moving at 6,000,000mph. Infinite speed is infinite speed. Relativity makes the invariant speed finite, and it turns out to be 3x10⁸ms^-1 (and hence forces the relativity of time and space - length contraction and so forth). It turns out that light travels at this invariant speed. That is not a coincidence, and there are good physical reasons for it. However, the sense Haushofer is using "speed of light" here is more closely related to the concept of it being an invariant speed than to the fact that light travels at the invariant speed.

Exactly how EM radiation would work in a Newtonian universe is unknown. The fact that our theories of EM radiation don't work in a Newtonian universe is how we ended up at relativity.

 

[PeroK]

Something that probably needs explaining here for the OP. Both Einstein and Newton envisage an "invariant speed", a speed that is exactly the same to all observers. In Newtonian relativity, that speed is infinity - if something passes me at infinite speed, it doesn't matter whether I consider myself to be stationary, moving at 60mph, or moving at 6,000,000mph. Infinite speed is infinite speed. Relativity makes the invariant speed finite, and it turns out to be 3x10⁸ms^-1 (and hence forces the relativity of time and space - length contraction and so forth). It turns out that light travels at this invariant speed. That is not a coincidence, and there are good physical reasons for it. However, the sense Haushofer is using "speed of light" here is more closely related to the concept of it being an invariant speed than to the fact that light travels at the invariant speed.

Exactly how EM radiation would work in a Newtonian universe is unknown. The fact that our theories of EM radiation don't work in a Newtonian universe is how we ended up at relativity.

I must say that I've never heard of Newton's believing that light has infinite speed (that's a mathematical contradiction to begin with). It's only Wikipedia, but the reference I have is that:

The first quantitative estimate of the speed of light was made in 1676 by Rømer. From the observation that the periods of Jupiter's innermost moon Io appeared to be shorter when the Earth was approaching Jupiter than when receding from it, he concluded that light travels at a finite speed, and estimated that it takes light 22 minutes to cross the diameter of Earth's orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth's orbit to obtain an estimate of speed of light of 220000 km/s, 26% lower than the actual value.[113]

In his 1704 book Opticks, Isaac Newton reported Rømer's calculations of the finite speed of light and gave a value of "seven or eight minutes" for the time taken for light to travel from the Sun to the Earth (the modern value is 8 minutes 19 seconds).

https://en.wikipedia.org/wiki/Speed_of_light#First_measurement_attempts

Newton, therefore, was very well aware of the finiteness of the speed of light and had a fairly good estimate of its value.

What Newton's mechanics assumes is that the speed of light would vary between moving reference frames. It was the invariance of the speed of light (not its finiteness) that was a problem for Newtonian mechanics.

Newton, however, as far as I am aware, had no explanation for how gravity could instantaneously propagate itself. His theory of gravitation, therefore, relied on instantaneous communication of the position of two masses with respect to each other.

 

[Dale]

I'd say that in both relativity-theories inertial observers measure the same speed of light. However, in Newtonian/Galilean relativity this speed is infinite (hence the appearance of an absolute time) and in Einstein's theory this speed is finite. I think that's the crucial difference.

This is very confusing.

Under the Galilean transform the invariant speed is infinite, and the measured speed of light is not equal to it.

Under the Lorentz transform the invariant speed is finite, and the measured speed of light is equal to it.

 

[Dale]

Newtonian force (F=dp/dt) doesn't need to be modified to be compatible with relativity.

Unmodified Newtonian force can easily accelerate to v>c, so you have to make modifications to prevent that. You can hide the modifications inside F or p, but either way you have to modify Newton's laws.

 

[Ibix]

I must say that I've never heard of Newton's believing that light has infinite speed

A snappier way to write what I wrote: you can talk about the speed of light in two senses - either as "the invariant speed (at which light happens to travel)" or as "the speed at which light travels (which happens to be invariant)". Since haushofer knows his stuff, I take it he's talking in the first sense and incidentally carrying the parenthetical statement with it (probably unjustifiably, as you point out).

 

[robphy]

This thread may provide background to haushofer's comment
https://www.physicsforums.com/threads/the-speed-of-light-and-of-galilean-relativity-comments.849242/
(I made a few comments there)

 

[haushofer]

This is very confusing.

Under the Galilean transform the invariant speed is infinite, and the measured speed of light is not equal to it.

No. But we also measure gravitational waves, while in Newtonian gravity these don't exist. The difference between the measured value of c and the limit c --> oo are post-Newtonian effects. So neglecting these effects is similar to treating c as inifinite.

I'd say it is not confusing when one reason from the underlying symmetry groups.

 

[haushofer]

A snappier way to write what I wrote: you can talk about the speed of light in two senses - either as "the invariant speed (at which light happens to travel)" or as "the speed at which light travels (which happens to be invariant)". Since haushofer knows his stuff, I take it he's talking in the first sense and incidentally carrying the parenthetical statement with it (probably unjustifiably, as you point out).

Yes. I like to think about this stuff as in effective field theories. In the standard model we also probably (:P ) neglect physics beyond a certain energy scale, say particles much heavier than a certain parameter E. Renormalization is then just sending E to infinity and decoupling physics beyond E from physics up to E, i.e. we pretend these heavier particles we don't know yet can not contribute to our path integral. I regard the c --> oo limit of special relativistic field theories likewise; high energy ('special relativistic') processes like particle creation, propagation of EM-waves and time dilation are decoupled and thrown away.

But I admit one has to be careful in distinguishing between c as some sort of parameter which can be contracted (in the underlying Lie algebra or eqn's of motion) and the measured value in experiments. So the phrase "c is infinite in Galilean/Newtonian relativity is mend (by me) as a limiting procedure on the parameter c, certainly not as a historical claim!

 

[DrStupid]

Unmodified Newtonian force can easily accelerate to v>c, so you have to make modifications to prevent that. You can hide the modifications inside F or p, but either way you have to modify Newton's laws.

No, that doesn't require a modification of Newton's laws of motion. Acceleration above c is already prevented by Lorentz transformation.

 

[David Lewis]

You can talk about the speed of light in two senses - either as "the invariant speed (at which light happens to travel)" or as "the speed at which light travels...

You've nicely highlighted a crucial distinction here, which may apply to other branches of physics.
In the first sense, c is a property of space. If the universe contained no light, c would still be the same.

 

[Ibix]

Yes. I like to think about this stuff as in effective field theories. In the standard model we also probably (:P ) neglect physics beyond a certain energy scale, say particles much heavier than a certain parameter E. Renormalization is then just sending E to infinity and decoupling physics beyond E from physics up to E, i.e. we pretend these heavier particles we don't know yet can not contribute to our path integral. I regard the c --> oo limit of special relativistic field theories likewise; high energy ('special relativistic') processes like particle creation, propagation of EM-waves and time dilation are decoupled and thrown away.

But I admit one has to be careful in distinguishing between c as some sort of parameter which can be contracted (in the underlying Lie algebra or eqn's of motion) and the measured value in experiments. So the phrase "c is infinite in Galilean/Newtonian relativity is mend (by me) as a limiting procedure on the parameter c, certainly not as a historical claim!

That's an interestingly different perspective from mine. I guess we really need to define what we mean by "Newtonian physics". Do we mean the historical situation where we genuinely believe Newtonian mechanics to be an accurate description of the world, and things like Maxwell's equations (and hence the speed of light) are puzzle pieces that don't quite fit? Or do we mean what we get if we start with a modern understanding and make formal approximations in the appropriate limits?

In the latter case, attempts to measure the speed of light are presumably (?) out-of-bounds because we've formally stated that $v<<c$ and we therefore can't play around near $c$. In the former case attempting to measure $c$ will eventually force us into a re-examination of the accuracy of Newtonian mechanics.

Anyway - I suspect this is a bit much for the OP, if he's still reading.

 

[Dale]

No, that doesn't require a modification of Newton's laws of motion. Acceleration above c is already prevented by Lorentz transformation.

The Lorentz transformation is not compatible with Newton's laws, that is precisely the point.

 

[Dale]

No. But we also measure gravitational waves, while in Newtonian gravity these don't exist. The difference between the measured value of c and the limit c --> oo are post-Newtonian effects. So neglecting these effects is similar to treating c as inifinite.

I'd say it is not confusing when one reason from the underlying symmetry groups.

It is confusing because you are using the phrase "the speed of light" to refer to "the invariant speed" rather than to "the speed of electromagnetic waves".

The terminology is standard, but it is inherently confusing, and you are making it more confusing by not acknowledging it but rather pretending like a novice should be aware of things like underlying symmetry groups. The poor OP asked for responses at a "B" level and not only are you not targeting your responses to such a level you are actively making it more difficult for those who are.

 

[PeterDonis]

Newtonian force (F=dp/dt) doesn't need to be modified to be compatible with relativity.

Yes, it does; as Dale pointed out in a previous post, it gets changed to $F = dP / d\tau$, where $F$ and $P$ are now 4-vectors and $\tau$ is proper time, not coordinate time.

 

[vanhees71]

The confusion comes from looking at the historical development rather than the structure of physical theories. Einstein's breakthrough in 1905 was not the mathematics of the Lorentz transformation, which is due to Voigt in the late 1880ies and FitzGerald, Poincare, and Lorentz in the early 1900s, but his discovery that the "speed of light" is a universal natural constant defining the mathematical structure of the description of space and time. This was finally understood completely in 1908 by Minkowski who formulated the space-time symmetry as a pseudo-Euclidean affine manifold.

From this perspective, it's more natural to introduce SRT by analyzing the symmetries of a space-time manifold, where there exist (global) inertial reference frames, such that for any inertial observer space and time are homogeneous and space is Euclidean. This analysis leads, up to equivalence, to two possible mathematical structures, namely Galilei-Newtonian spacetime with the (inhomogeneous) Galilei group as the symmetry group or Einstein-Minkowski spacetime with the Poincare group as the symmetry group. In the latter case a universal "limiting speed" is introduced, which governs all physical laws since it's part of the underlying spacetime structure.

Whether or not the speed of light (i.e., the phase velocity of electromagnetic waves in vacuo) is equal to this limiting speed then is an empirical question. In a modern way one can formulate it as the question, whether the electromagnetic field is really exactly massless. Today the upper limit of the photon mass is $m_{\gamma}<10^{-18} \mathrm{eV}/c^2$. In the Standard Model of elementary particle physics we take the photon as exactly massless.

 

[Ilja]

In fact, Poincare was quite close in the spirit to the Minkowski interpretation even years before 1905 (but, sorry, don't ask me to provide quotes, this is out of memory of my youth). So one can find quotes of Poincare such what it is hard to find a difference to the spacetime interpretation. The guy who believed that only one of the many mathematically equivalent frames is the true rest frame of the ether was Lorentz, so it is quite correct to name this interpretation the Lorentz ether.

On the other hand, one can also make a case that Poincare has not transcended the limits of classical spacetime philosophy. I would summarize this as that he was thinking in the direction of spacetime interpretation but had enough philosophical background not to take these ideas too seriously. It is, essentially, one thing to recognize that I cannot decide, by observation, which of two theories (say about which frame is the true rest frame) are true, and the idea that it follows that there is no such truth, that above theories are equally valid.

 

[Grimble]

Thank you Gentlemen! Yes I am still reading.
I am limiting myself to Special Relativity as there seems little point in studying and trying to understand GR until I understand SR.

I like Occam's approach and like things to fit without assumptions, or guesses.

Surely the difference between Newton and Relativity is not about whether Newton thought the speed of light was infinite - as has been pointed out, he didn't, but rather, that it was not a general limit to the speed of everything.

Is the rotation of a moving Frame of Reference not the rotation of the time axis due to the movement of the traveling frame?

I have come to think the invariance of the speed of light to be easily explained:
light travels between two events at c; the speed of light in gravity free space.
Those two events exist in every Frame of Reference; only their coordinates differ.
Every observer is at rest in their own local Frame of Reference, therefore the locations of those two events are fixed for each and every Frame...
Therefore the locations of the two events are fixed in every frame - locations of events cannot move...
Therefore light will travel between the same two locations in the same time - proper time (as that is the time measured locally at the origin of any inertial FoR.
Therefore light, must always travel at the same speed - relative to Spacetime - in every FoR; which is how the speed of light is measured - it would be pointless to try and measure the speed of light relative to anybody in space, wouldn't it?

That just seems to me to be the simple straightforward way to view it...

Also I understood that at least much of Newtonian mechanics (apart from the speed of light) was true as a limiting case that it only differs as speed increases?

 

[PeroK]

I have come to think the invariance of the speed of light to be easily explained:
light travels between two events at c; the speed of light in gravity free space.
Those two events exist in every Frame of Reference; only their coordinates differ.
Every observer is at rest in their own local Frame of Reference, therefore the locations of those two events are fixed for each and every Frame...
Therefore the locations of the two events are fixed in every frame - locations of events cannot move...
Therefore light will travel between the same two locations in the same time - proper time (as that is the time measured locally at the origin of any inertial FoR.
Therefore light, must always travel at the same speed - relative to Spacetime - in every FoR; which is how the speed of light is measured - it would be pointless to try and measure the speed of light relative to anybody in space, wouldn't it?

That just seems to me to be the simple straightforward way to view it...

So many words, so little maths!

What you have written may or may not makes sense. I suspect you've used words to hide the lack of mathematical consistency in your argument. You're obviously clever, but you need to hone your methods and thought processes.

 

[Dale]

Surely the difference between Newton and Relativity is not about whether Newton thought the speed of light was infinite - as has been pointed out, he didn't, but rather, that it was not a general limit to the speed of everything.

Yes, that is what we were describing above with the discussion of the invariant speed.

proper time (as that is the time measured locally at the origin of any inertial FoR.

Proper time is defined along a particular worldline and is the time on a clock moving on that worldline. Reference frames and origins aren't part of it.

 

[Grimble]

Proper time is defined along a particular worldline and is the time on a clock moving on that worldline. Reference frames and origins aren't part of it.

Yes, I understand that.
But can you explain to me how the time axis for a Frame of Reference is not the World-line of the substantial point (to quote Minkowski) that is at the origin of the Frame of Reference?
A Frame of Reference is no more than a map of Spacetime based upon that origin. It is, according to that map, that Frame of Reference, at rest and the time on a clock residing at the origin of the Frame of Reference is one moving along that worldline, which must consist of a single location in that substantial point's (and that imaginary Clock's) Frame of Reference the only coordinate changing through the life of that point or clock being the time coordinate - in that Frame of Reference.

Given that reasoning, how can the origin of a Frame of Reference not be the Worldline of the origin of the Frame of Reference and how could that clock keep any time other than Proper Time? - (It is after all "doing exactly what it says on the Tin" )

Or is there some specific Frame of Reference that a worldline has to be plotted on? But then there is no preferred Frame of Reference; is there? So that would only leave the concept of some sort of divine view of spacetime...

Let me assure you, Dale, for I have great respect for you, I am not trying to rewrite relativity, or how it is understood, but only pulling bits together and reasoning (to try and understand better) how they must work. (Isn't that what scientists are supposed to do?)

 

[Ibix]

I think you are trying to understand this backwards, @Grimble. You aren't wrong, but I think you are heading in a confusing direction.

All clocks show proper time. "Proper" in this context is coming from the same root as "property", something which is the clock's own. In the twin paradox, for example, there are two clocks. One shows ten years elapsed, the other shows twenty. That means that along one path the proper time was ten years and along the other it was twenty years.

You are correct that the proper time shown by a clock at rest in some reference frame is the same as the coordinate time in that frame. But saying a clock shows proper time is a tautology.

You compared a frame of reference to a map. This is right - and coordinate time and the spatial coordinates are like grid references on that map. Proper time is the distance along some arbitrary (possibly non-straight) line on the map (strictly I should say a space-like line). If that line is a straight line representing an object at rest in some frame then the proper time along the worldline is the same as the coordinate time at that location, true. But that's because we defined coordinate time as the proper time of a set of clocks at relative rest.

Hope that makes sense.

 

[PeterDonis]

can you explain to me how the time axis for a Frame of Reference is not the World-line of the substantial point (to quote Minkowski) that is at the origin of the Frame of Reference?

It can be, but it doesn't have to be, any more than the Earth's equator or prime meridian has to have actual markers all along it.

 

[Grimble]

It can be, but it doesn't have to be, any more than the Earth's equator or prime meridian has to have actual markers all along it.

I am sorry but I don't see how it can not be the worldline of the origin of the frame of Reference - In that map of spacetime it is at rest and therefore only the time will change - isn't the worldline just the path that the subject takes? So i a Frame of Reference how can the origin be any other than the worldline and the clock be any other than proper time?

 

[Dale]

But can you explain to me how the time axis for a Frame of Reference is not the World-line of the substantial point (to quote Minkowski) that is at the origin of the Frame of Reference?

A frame of reference is a mathematical object, not a physical object. In GR it is formalized as something called a tetrad, in SR it is less formally used to refer to a coordinate system. Either way it is a mathematical object, not a physical object. A reference frame is part of the analysis, not part of the experiment.

You can use a frame of reference regardless of whether or not there is any substance located at the origin. Many times it is convenient to use reference frames where no object is at rest (e.g. the center of momentum frame) so no object is located at the origin other than "in passing".

From a theoretical standpoint proper time is the spacetime interval along a given timelike worldline, which is experimentally measured by a clock traveling on that given worldline. Neither the theoretical nor the experimental meaning of proper time mentions a frame of reference nor even "inertial".

 

[the_emi_guy]

I am sorry but I don't see how it can not be the worldline of the origin of the frame of Reference - In that map of spacetime it is at rest and therefore only the time will change - isn't the worldline just the path that the subject takes? So i a Frame of Reference how can the origin be any other than the worldline and the clock be any other than proper time?

Grimble,
The fundamental difference between Newton and special relativity is quite easy for someone at your level to grasp. Some of the previous posts may be delving into the consequences of relativity which can get very complicated indeed. (Newtonian mechanics itself can get very complicated too...principal of least action, calculus of variations etc etc.).

The postulate of SR is that the speed of light is constant in all frames of reference. Let's explore in a very simple way what this means.

The only equation we need is d = (v)(t) (distance = velocity x time) and some high school geometry.

Consider that you are inside a train that is moving along at 1m/s. You are standing up holding a ball 1 meter above the floor. Next you move the ball toward the floor at a constant velocity of 1m/s. One second later the ball has reached the floor.

What do you observe? The ball moved vertically 1 meter and it took 1 second to do so. d = (v)(t) --> 1meter = (1m/s)(1s).

What does someone outside the train observe? During the one second that the ball was in motion, it moved 1 meter vertically and 1 meter horizontally due to the train's motion, total distance moved is 1.414 meters (Pythagoras). It also moved faster than 1m/s since it has both vertical and horizontal motion, velocity= 1.414m/s (by vector addition of horizontal and vertical velocities). d = (v)(t) --> 1.414m = (1.414m/s)(1s).

So these two observers disagree about how far the ball has moved, and how fast it moved. They agree on is how long it took the ball to get to the floor (1s).

The same would apply if the moving object were a wave. Back inside the train you pick up a 1m long vertical pipe and launch a sound wave which travels within the pipe, using the air in the pipe as its medium. Within the train, you will see a vertically moving sound wave moving 1 meter. The observer outside will see the soundwave moving faster since the medium (air in the pipe) has an additional horizontal component to its velocity.

Finally, consider a similar experiment with a flashlight held 1 meter above the floor of the train. You switch on a flashlight and observe the light traveling toward the floor. After 3.3nanoseconds the light will have moved down 1 meter to the floor. What does the outside observer see? The outside observer sees the lightwave travel a distance greater than 1 meter (the lightwave has a horizontal component to its motion due the motion of the train) and, according to Newton, the lightwave must have traveled faster in order to cover this longer distance during the same 1 second interval, just as was the case with the ball and soundwave.

However, according to Einstein, both observes must see the light traveling at the same velocity.

So, if the outside observer sees the lightwave travel a longer distance, but traveling at the same exact velocity, then according to d=(v)(t) the outside observer sees the lightwave take longer than 1s to get to the floor.

Thus the the two observers disagree about the how far the lightwave moved, and the amount of time it took to get to the floor, but agree on how fast it was traveling.

This is the "root cause" of all of the strange, counter-intuitive stuff that happens in SR: Two observers are moving relative to each disagreeing about the amount of time that elapses between events because they are forced to agree about the speed of the lightwave.

You can actually derive the Lorentz transformation directly using nothing more that the above thought experiment and the Pythagorean theorem
- try it, I think it would be a good exercise for you.

Cheers

 

[Grimble]

You can actually derive the Lorentz transformation directly using nothing more that the above thought experiment and the Pythagorean theorem
- try it, I think it would be a good exercise for you.

Thank you. I have done that and it does work very simply.
https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Mechanics%20compared.png?attachauth=ANoY7cp6yoIyeGFRapn5Qs4B7duBpjmEDFM5fzI_DLZ-ERtGRvW9pDP-om9pcekWU2pMUTBqFBS3BFVouGwnsy3D3f81IZMR_1DRalDIy449NhNH3ihzUJuRU4fzd18RpBmoqtqKMfGEf0JsFgvrsck_dec7Hh75CWJPOY0Hep5hcTbPsHQdRsmcRC521qCuFgVDxXkT_YasVP2V6djmjaTXoSaQq2U0FcLhRND1T1coizy59L12HvZMAN2i-S1fOhIi4LyjlI8jAqsZqQkRm8MTI7z_2y9YuA%3D%3D&attredirects=0

In Newtonian Mechanics the path of the light in Clock B measured from Clock A lies at angle α and travels 1.166ls in 1 second.
In Relativistic Mechanics - the second diagram - the light in Clock A, measured from Clock B, as we know travels 1ls in 1 second; while also traveling 0.6ls horizontally from Clock B. Thus arriving at point (-0.6,0.8) after 1 second, giving us a different path angle β.

The important difference I see here is that in Newtonian Mechanics it is tan α = vt/ct but in Relativistic Mechanics it is sin β = vt/ct.

That is the effect of the invariance of c.

And as you say the second diagram gives us the time measured for the light to reach the mirror in a resting clock = 1 second, the time in a clock moving at 0.6c = 1.25 seconds so ct = √ ( (ct)'² - (vt')² )= ct'√ ( 1-v/c² )
where c = 1,
v = 0.6
this gives t = t'√ ( 1-v/c² ) or t'/γ

t = 0.8 t' or t' = 1.25 t which we see in diagram 2 and the Lorentz Factor for 0.6c is 1.25

 

[the_emi_guy]

Thank you. I have done that and it does work very simply.

Nice job, I'm impressed!
Nice diagrams too!

Next to consider is that even though the observer on the platform measured 1.25 seconds (his "coordinate time"), he is not oblivious to how long the the observer in the train measured. Using the time he measured (hypotenuse, 1.25sec) and the velocity of the train, he can work Pythagorean backwards and compute the time elapsed on train observer:
1.25² - 0.75² = (train observers time)² = 1²

We call this the "proper time" \tau
Δ\tau² = Δt² - (Δx/c)²

Proper time is a useful concept because it is "invariant", all observers will agree on the "proper time" between events irrespective of their relative motion.

Armed with this insight, I recommend the following YouTube video:



Cheers

 

[Ilja]

All clocks show proper time. "Proper" in this context is coming from the same root as "property", something which is the clock's own.

Yes, the original German term is "Eigenzeit". The word "eigen" translates also as "private", "own", "separate", "distinctive". I think every of these translations would have been more appropriate than "proper", which has also the association of true, correct, genuine, appropriate, adequate. Associations which are misleading, and not present in the German "eigen", which also has "peculiar" as a possible translation. http://www.dict.cc/?s=eigen

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.