# How do mechanical processes depend from the speed of light?

1. Apr 17, 2012

### Aidyan

For example, as well known the period of the pendulum is (in linear approximation):

$T \approx 2\pi \sqrt\frac{L}{g} \,.$

So, no speed of light appears explicitly. What I'm wondering however is if and how it might be implicit? In the sense that after all the tension in the rod depends from molecular forces, which at the microscopic scale are of electric nature. Would a different speed of light than c lead to a different length? Another purely mechanical example: hitting a nail in a wall. In a universe with, say light speed 0.7*c, would it become more or less easier to hit the same nail in the same wall? These are only an example of a more general question. In what way is classical non relativistic mechanics determined by the speed of light, if it does?

2. Apr 17, 2012

### Naty1

In classical mechanics 'g' is simply the acceleration of any freely falling body....a variable itself....and that's the assumption in your pendulum formula. Whether the speed of light is 'c' or 2c or or 1/2c whatever has virtually no effect on a clock pendulum.

It's analogous to the simple addition of two velocities at low speeds.....which works fine....rather than the 'exact solution'....As you approach speeds which are a significant proportion of 'c' however, you need to take finer grained details into account...that's relativistic mechanics.

I'm not exactly sure about how the electromagnetic force that binds electons to nuclei, or the strong force that binds atom nuclei together, for example, might vary if the speeds of propagation were to vary from 'c'....

3. Apr 17, 2012

### AlephZero

Non relativistic mechanics ignores the speed of light.

That is usually a very good approximation, cosidering that if your mathematical description of a pendulum, knocking in a nail, or whatever, includes the elasticity of the material, any mechanical effect only propagates through the body at the speed of sound in the material (typically a few km/s for metals), not at the speed of light.

4. Apr 17, 2012

### Aidyan

I'm not convinced. Also in classical mechanics the elasticity of the material depends somehow from the molecular and atomic lattice structure and which solidity in turn is determined by the inter-atomic bonds. These bonds are of electromagnetic nature and therefore I imagine them dependent from c. So the speed of sound must depend from c too.

5. Apr 17, 2012

### AlephZero

You can imagine whatever you like, but that doesn't make it so.

6. Apr 17, 2012

### sophiecentaur

There is a logic in what he says, surely: in principle, even if it's not quantitatively very significant. The time taken for molecules to 'rebound' off each other will depend upon the photon interaction between them. But the effective difference between that and 'billiard balls' collisions may be slight. The velocities of sound and light differ by a factor of around a million.

7. Apr 17, 2012

### rbj

$c$ is not simply the speed of light (or of the electromagnetic interaction). it is the speed of any ostensible "instantaneous" interaction. including the strong nuclear force and including gravitation.

so, if the question is: "why does a mechanical clock, instead of a light clock, appear to slow down due to time dilation as it whizzes by an observer at high speed?" then the answer is that the constancy of $c$ does not only cause time dilation, it affects length contraction and apparent momentum of mechanical parts.

8. Apr 18, 2012

### Aidyan

Ok, that might be true in an idealized gas of free particles, but reality is a bit more complex. Especially in a solid lattice structure (e.g. a metal rod for the pendulum example) the resistance to deformation (and therefore its length variation due to gravitational pull) is determined by the chemical bonds. In a simplified model we can visualize a chemical bond between two point masses with a spring governed by Hook's law: F=-k*dx, where dx is the displacement from force equilibrium, and k the Hook's constant. This sounds all very mechanical an independent from the speed of light. But truth is that a chemical bonding (ionic or covalent) is determined by the electron density, bond length, its energetic stabilization, etc., all electric phenomena which in turn must dependent from c somehow. That is, further analysis must show that k=k(c). The question is not if this is the case but to what degree it is. I suspect that the bond strengths depends significantly from the value of the speed of light, and therefore also all the other mechanical properties. I looked up for some formula on the web but couldn't find much.

Yes, precisely. A reason more to suspect that also "g" in the pendulum formula must depend from the speed of light.

As far as I understand SR everything slows down, also biological aging.
Anyhow, rbj, I was expecting you to tell us that since c is a dimensional constant the question then is meaningless, since nothing would change if the speed of light changes....

Last edited: Apr 18, 2012
9. Apr 18, 2012

### sophiecentaur

Precisely. It's always more complicated than that. One of the reasons is that structures rely on photons to keep them stiff and gases rely on photons to establish 'pressure'. Any photon interaction takes time,which depends upon c.
But, there again,everything 'depends on c'.

10. Apr 18, 2012

### Staff: Mentor

If you change c, but not anything else (in SI units), you change the fine-structure constant and therefore the strength of the electromagnetic interaction, which changes the binding energy and size of atoms. While this can be absorbed with scalings of length and energy for small atoms, for large atoms you mess around with the fine-structure and might get a different periodic system, if the modification is large enough.

Your ideal gas will not change in a significant way, but a solid or liquid material can change its properties a lot.

11. Apr 18, 2012

### sophiecentaur

If an atom comes across another atom, it will only 'bounce off' because a a photon interaction of some kind. (It's not gravitational or nuclear forces at work). The system we would be dealing with would not just be one atom any more. I guess you could call it a "fine structure" effect because the energy levels of each atom would be modified by the proximity of the other atom. As you say, it would be a very small effect but the same basic thing would apply with two isolated atoms or with a set of inter-linked atoms. It would still take d/c time for the electric forces to start to work. (d being some effective distance)

Questions like this one sometimes end up with needing to go to unreasonable depths to get a result that satisfies everyone because we all have our own basic model in our heads.

The idea of 'changing c' is really a bit of a non-starter because everything hangs on it. However, you can say that many problems can be solved quite satisfactorily without considering it. That goes for most 'mechanical' ones.

12. Apr 18, 2012

### Staff: Mentor

This bounce between gas atoms/molecules is elastic and does not need any time in an ideal gas, therefore the details of the bounce do not matter. It could be any force, even gravity.

13. Apr 18, 2012

### sophiecentaur

Umm. Just becaust no energy is lost, it doesn't mean that the time taken is zero or irrelevant. You have introduced a mechanical concept so I will reply with one. The time constant of the spring/mass system would have an effect on the bounce time.

14. Apr 18, 2012

### Bob S

15. Apr 18, 2012

### Staff: Mentor

Therefore, I highlighted ideal gas. The ideal thing about it is that the interactions are assumed to be small, elastic and do not require any time.
A real gas will change its properties of course. But gases are much more "robust" in that sense than solids or liquids.

16. Apr 18, 2012

### Aidyan

That's precisely the thing that isn't sure at all. Please read wiki on Planck units, paragraph "Planck units and the invariant scaling of nature".

17. Apr 18, 2012

### sophiecentaur

In practice but not in principle (or do I mean in principle not in practice?) - as an 'ideal' gas doesn't exist. The behaviour of any real gas will be a bit of a slave to c so we are only arguing about a matter of degree.