# B Newtonian vs Relativistic Mechanics

1. Apr 1, 2016

### 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

2. Apr 1, 2016

3. Apr 1, 2016

### 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.

4. Apr 1, 2016

### Staff: Mentor

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.

5. Apr 1, 2016

### 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.

6. Apr 1, 2016

### DrStupid

Newton: Galilean transformation
Einstein: Lorentz transformation

Everything else in SR results from this difference.

7. Apr 1, 2016

### 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:
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?

8. Apr 1, 2016

### DrStupid

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.

9. Apr 1, 2016

### Ibix

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.

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 realise 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.)

10. Apr 1, 2016

### PeroK

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.

11. Apr 1, 2016

### Staff: Mentor

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.

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)

12. Apr 1, 2016

### 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.

13. Apr 1, 2016

### PeroK

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".

14. Apr 1, 2016

### Ibix

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.

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...

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.

15. Apr 1, 2016

### Staff: Mentor

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

16. Apr 1, 2016

### pervect

Staff Emeritus
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 lets 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.

17. Apr 2, 2016

### Staff: Mentor

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.

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

18. Apr 2, 2016

### Grimble

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 travelled 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 travelled 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?)

19. Apr 2, 2016

### PeroK

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.

20. Apr 2, 2016

### Staff: Mentor

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

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.

Last edited: Apr 2, 2016