Are Physics Equations Always Exact?

[Nick89]

Hi,

I have been wondering about this for some time and thought I would ask here...
It may be a stupid question but I really don't know...

I was wondering if the equations that allow you to calculate things in physics are exactly true, or merely true within a certain error?

For example, take coulomb's law to calculate an electric field:
$$E = k \frac{ q_1 q_2 }{r^2}$$

Is there any 'proof' that the '2' in $r^2$ is exactly a 2? Would it be possible that it's actually $r^{2.00000017858726}$ or something?

I understand these things can be verified by doing experiments, but those will tell you the answer to 'only a few' decimals... If experiments verify that the 2 is indeed 2.0000000000, maybe it is infact 2.000000000000000000000018358...

I hope you understand my point...

 

[HallsofIvy]

It's not so much the equations as the fact that the measurements you put into the equations that are not exact.

As far as the 1/r² in electric, magnetic, and gravity are concerned, it's pretty easy to show that anytime a specific quantity spreads out evenly in space, it must obey a 1/r² law in order to keep the same total quantity. If the formula involves r to a power, then that power must be -2.

Of course, in general relativity, the formula is nothing like that.

 

[Cvan]

Considering the limit to which things are experimentally measurable (and the fundamental uncertainty involved in a particle's position//velocity, etc), who is to say that after a certain point any 'integer' quantity specified by man is even relevant?

All things in life are only true within a certain degree of error.
As my cosmology teacher used to say--No error bar, no physics.
It's not necessarily that the equations are wrong, but there is a fundamental limit to observation as we know it.

 

[Nick89]

So when you start looking at the extremely small 'error' the 2 may or may not have, you are essentially looking at quantum physics? (I don't know much about that so..)

And about the evenly spreading quantity, that makes sense yes. But there must be some other example (can't think of one right now...) where it isn't directly clear, no?

I understand that in GR the formula is going to change (although I also don't know very much about that), does this mean the formula is essentially wrong, or is it merely only right for specific circumstances?
I think what I'm trying to ask is, are formulas like these always valid or do you find that they are no longer valid when looking on extremely small or large scales for example?

 

[rbj]

Hi,

I have been wondering about this for some time and thought I would ask here...
It may be a stupid question but I really don't know...

I was wondering if the equations that allow you to calculate things in physics are exactly true, or merely true within a certain error?

For example, take coulomb's law to calculate an electric field:
$$E = k \frac{ q_1 q_2 }{r^2}$$

Is there any 'proof' that the '2' in $r^2$ is exactly a 2? Would it be possible that it's actually $r^{2.00000017858726}$ or something?

I understand these things can be verified by doing experiments, but those will tell you the answer to 'only a few' decimals... If experiments verify that the 2 is indeed 2.0000000000, maybe it is infact 2.000000000000000000000018358...

I hope you understand my point...

besides experimental verification, there is a solid theoretical basis for the 2 in the denominator exponent in inverse-square laws. it has to do with the concept of flux and flux density, conservation of this physical quantity, and that the surface area of a sphere in 3-dim space is $4 \pi r^2$.

 

[Nick89]

I think I understand that now yes.

However, can you explain how it's then possible to also have magnetic / electric fields that obey a $r^{-1} \text{ or } r^{-3}$ law?
I would just like to know, I think it's interesting...

 

[rcgldr]

[QUOTEIAs far as the 1/r² in electric, magnetic, and gravity are concerned, it's pretty easy to show that anytime a specific quantity spreads out evenly in space, it must obey a 1/r² law in order to keep the same total quantity. If the formula involves r to a power, then that power must be -2.[/QUOTE]

exception?

Strong force is an exception, although I'm not sure it's the same kind of "force" as the others mentioned above.

http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html

http://aether.lbl.gov/elements/stellar/strong/strong.html

For the forces listed above, the 1/r² is exact, it's only the constants that are derived or calculated from experiments and/or measured. 1/r² relationship is exact for a point source or sphere. 1/r is the relationship if the force eminates from an infinitely long line (or cylinder), and force is a constant if it eminates from an infinitely large plane.

 

[benabean]

some possible help here

the 'squared' part in the inverse law equations has something to do with being along the lines of 'to the power of (no. of dimensions - 1)' ie 3 dimensions leads to the 'squared' part.

 

[chuy]

Well, r^2 actually could be r^2.00000123... , we don't know, until we taste it with experiments. But then, 2 is very accurate. When we do models and theories we must be aware that our models might be not fully exact, but if these give some good results, and are accurate within their own range of validity, then they are good models, and we can use it to explain things. But we don't know if these models are perfectly exact. Although we can tray to improve this even more.

 

[Nick89]

Well that was really my question, wether we only know the 2.000... to be correct within a certain error, or wether you can prove that it's exactly 2.

I thought of this when I realized that I have only ever seen 'neat' or powers in equations.

For example, it's quite common to see an equation like $y = ax^2$ where a is some constant we can 'only' approximate up to like 16 decimals.

However, I have never seen any equation like $y = 2x^a$, again where a is some 'dirty' constant. (For example, $y = 2x^{(-36.13986092...)}$)

Maybe I just have never encountered them. Are there any equations that are like that?

I just think it's fascinating that most physics equations are so 'simple', where the only numbers we can only approximate are usually just the proportionality constants (like G) or numbers like pi.

 

[chuy]

Well, there exist a lot of theoretical reason for why r^2, must be 2 and no 2.00031415..., one example is dimensional analysis. What does (meter)^2.000123... mean?

But as far as we are dealing with reality, we cannot prove things like mathematical proofs, and even nature can behave weird, and really 2.00031415... maight be the number that works (In fact we could prove that something is wrong, but we are not able to prove with 100% of certainty that something is completely right). But as long as we don't find a discordance with experiments in our models, these will continue being good models to explain the nature.

 

[rcgldr]

The parts based on math (like calculus) are exact, the parts based on measurements or experiements are just very close.

 

[lightarrow]

Is there any 'proof' that the '2' in $r^2$ is exactly a 2? Would it be possible that it's actually $r^{2.00000017858726}$ or something?

If that power wouldn't be 2 then the photon would have a non-zero mass. This is general, for any infinite-range interactions. For finite-range interactions, the mediator has non-zero mass (strong interaction, weak interaction, e.g.). The less the range of interaction, the grater the mediator's mass.
The photon's mass has been measured and found = 0 within the experimental limits. So, within those limits, the power value is exactly 2.

 

[dx]

No, no one knows whether the power is exactly 2, or if the formula is true at the smaller distances. On normal scales ( i.e distances of the order of one angstrom) it has been shown that the exponent differs from two by less than one part in a billion. Theres a good discussion of this in Feynman's lectures vol 2.

 

[Andy Resnick]

Hi,

I have been wondering about this for some time and thought I would ask here...
It may be a stupid question but I really don't know...

I was wondering if the equations that allow you to calculate things in physics are exactly true, or merely true within a certain error?

For example, take coulomb's law to calculate an electric field:
$$E = k \frac{ q_1 q_2 }{r^2}$$

Is there any 'proof' that the '2' in $r^2$ is exactly a 2? Would it be possible that it's actually $r^{2.00000017858726}$ or something?

I understand these things can be verified by doing experiments, but those will tell you the answer to 'only a few' decimals... If experiments verify that the 2 is indeed 2.0000000000, maybe it is infact 2.000000000000000000000018358...

I hope you understand my point...

People seem to have focused on your example, rather than your general question- maybe I can offer some perspective.

A model system is one we have invented, because certain aspects of the model correspond well with observations and the model omits many extraneous confounding effects. Many physical phenomena can be modeled by a mass-spring system, for example. Electrostatics, magnetostatics, etc can be modeled by vector fields and scalar potentials. I am modeling Autosomal Dominant Polycystic Kidney Disease by the culture of epithelial (cell) monolayers. Crystals can be modeled as periodic objects, etc. Models can be analytic, computational/numerical, or experimental. A wind tunnel is a good example of an experimental model.

What I think you are asking, is when a model gives an 'exact' result, does that mean that the actual real-life system *must* correspond exactly? The answer is no. But, a good model will give very prcise and accurate predictions. Given several different models, choosing a 'correct' model can be difficult, subject to debate and scientific investigation.

So, there is a good reason why the model of electrostatics gives an exponent of '2'- the observed number of dimensions, conservation of energy, an so on. However, there is no guaruntee that an infinite number of future experiments will never measure a value different that 2.0000000... . If someone did, that would mean the model (three-dimensional world, massless photons, conservation of energy..) is insufficient and requires modification.

 

[M Grandin]

Hardly just a constant added in the r-exponent

My professor in theoretical electromagnetics mentioned the possibility of other exponent than 2 in the distance formula long ago.

But if there is a slight difference from 2, it could hardly be just a constant added.

If 2 is not exactly true, it might depend on also the total flux is attenuated according to distance r. Resulting in for instance an inverse exponential factor 1 / e ^(const x r) added, resulting in
2 replaced by 2 + const x r / ln r . :grumpy:

 

[pam]

Some people take the question of how close to 1/r^2 Coulomb's law is quite seriously.
The most recent measurement I know of (PhysRevLett.v26,p721,(1971)) measures
1/r^(2+q), and finds q consistent with zero, with an upper limit of ~3X10^(-16).

 

[rbj]

I think I understand that now yes.

However, can you explain how it's then possible to also have magnetic / electric fields that obey a r ^-1 or r ^-3 law?
I would just like to know, I think it's interesting...

dunno about an inverse-cube law. (is the E field from off the end of a dipole inverse cube? i think it is.)

the answer to both lies with geometry and mathematics. the fields due to the constituent point charges are still inverse-square. given that fact, other geometries give different relationships:

dipole: r ^-3 at a large distance.
point charge: r ^-2
infinite line of charge: r ^-1
infinite plane of charge: r ^-0

 

[Count Iblis]

A theoretical viable way to get a violation of Coulomb's law is by giving the photon a small mass. The Colomb's law would change according to 1/r^2 --> exp(-r/c)/r.

The article mentioned by Pam gives an experimental limit on the devation of Coulomb's law and thus also a limit on the photon mass. It is possible to measure this devation accurately, because Gausss law depends on the validity of Coulomb's law. And, in turn, if Gauss's law is not exactly valid, you can have an electric field inside the cavity in a conductor if there is an electric field on the outside of the conductor.

Such an anomalous electric field can be measured very precisely. Null results of such experiments give you accurate boiunds on the photon mass.

Now, it is possible to deduce much more stringent limits on the photon mass, e.g. using the fact that if the photon had a mass then the vector potential itself would have detectable effects and not just the electric and magnetic fileds. The galactic verctor potential is numerically very large despite the fact that the galactic magnetic fields is very small because it extends over such a large distance (vector potential is of the order of magnetic field times radius).

However, there are some hidden assumptions in such an argument, in particular about the mechanism that gives rises to a hypothetical photon mass http://arxiv.org/abs/hep-ph/0306245" . Change your photon mas scenario and the photon mass bound vanishes.

 

[Nick89]

I see. Some interesting reads here!

However, (maybe it's just me misunderstanding slightly) I see a few contradictions here.

With the r^(-2) as example again, I see a few people saying the 2 comes from geometrical / mathematical analysis. If this is true, I am quite at peace with it being exactly 2.
However, others say the 2 may not be exactly 2, but it is within the limit of measurement. If it's not exactly 2, can it still be related to geometrical / mathematical results?

One way to maybe explain my remaining 'problem' is this.

Let's look at gravity for example: $$F = G \frac{m_1 m_2}{r^2}$$

In this equation, there are two significant (to my question) constants: G and 2.
The G is ofcourse a number we don't know exactly, since our measurements are limited to 'only a few' decimals. If we can measure better in the feature, we can determine G to more and more decimal places. This however will not alter the equation itself (will it?) Since the G as we use it in the equation is assumed exact (right?), we merely don't know the exact value of it as of now (if there even is an exact value, see later on).

The 2 however, seemed to me to be precisely accurate (I understand now that we can't know this for sure) since even with more precise measurement, the 2 will not change.
Also, any 'anomalies (sp?)' in the 2 cannot be corrected by simply specifying a constant like G to more decimal places (since it's in the exponent of a variable obviously..)Well, I'm not sure I understand fully yet but I do realize now we simply cannot be sure that the physics formulas are exact (which I expected from the start) because they are simply models. So even those things my teacher calls 'beautiful symmetry' (mentioned above for example with the electric fields, dipole: r^-3, point: r^-2, line: r^-1, plane: r^0) may not be so beautiful after all...
Finally, I have also been thinking about something related.
Do we know for sure that the 'proportionality constants' like G are irrational? Can this be (dis)proven?

What if G was rational, for example 1.23456 (I know this is way off, just an example), so no more numbers follow (1.234560000000000000...)

Couldn't we then simply change our units to make G exactly 1 which would make the equations even simpler?

Ofcourse doing this would then obviously make other constants 'ugly' again so there wouldn't be much gained... lol...

 

[Count Iblis]

Nick:

Couldn't we then simply change our units to make G exactly 1 which would make the equations even simpler?

That's a very clever observation. You can indeed do that. In theoretical physics people will often choose a set of units such that there are no dimensional constants.

 

[DavidWhitbeck]

Yeah in GR geometric units are usually used where $$G=1=c$$.

 

[Nick89]

But isn't that only possible if G is rational? Maybe I'm overlooking some way but to me the easiest way to make G = 1 would be for example:

Assume G = 1.234 (standard units)
Multiply by 1000: G = 1234 (arbitrary units)
Divide by 1234: G = 1 (arbitrary units)

If G is irrational, this obviously doesn't work
Assume G = 1.2345...
Multiply by... What? Infinity? :p

 

[DavidWhitbeck]

But isn't that only possible if G is rational?

With respect to what units system? There is no absolute unit system and saying that $$G=1$$ amounts to a choice in units.

 

[rbj]

me piling on...

Couldn't we then simply change our units to make G exactly 1 which would make the equations even simpler?

Ofcourse doing this would then obviously make other constants 'ugly' again so there wouldn't be much gained... lol...

but there are at least 3 degrees of freedom in our choice of units. we can (and have) arbitrarily choose our unit length, unit mass, and unit time. they are currently arbitrarily chosen in the SI system: 40,000,000 meters was, at one time, s'posed to be the circumference of the Earth. once the meter (and cm) are defined, a cube of 10 cm on a side (a.k.a. a "liter") of water was s'pose to be a kilogram, and the second has always been with us (i don't know the history of why or when the hour was divided into 60 minutes which each were divided into 60 seconds) 1/86400^th day. those are pretty arbitrary definitions. almost as bad as measuring the distance from the king's nose to his fingertip.

anyway, with those 3 base units pre-defined, then human beings went out to measure G, c, and $\hbar$. but we could instead define the units length, mass, and time to be "variable" (or better yet, "unknowns"), set all three G, c, and $\hbar$ to 1 (not so ugly) and from those 3 constraints, solve for these unknown units in terms of the SI defined units. they would be called Planck units and you can read about such online.

now the ugly constants that never go away are the dimensionless ones (pure numbers with no units attached). and those dimensionless constants that are parameters of the universe and the only ones we think are fundamental.

 

[Nick89]

Yeah that sounds understandable enough. I know most of our units are just arbitrarily chosen and have been changed many times*, I just didn't figure out this allowed us to make important constants equal...

*It's correct that the meter is now (since the speed of light has been defined exactly) defined to be the length that light travels in 1/c seconds, right?
And the second is defined as some large number of some transitions in the hydrogen atom, right?

Hasn't this essentially changed the length of one meter?

Let's say you have a stick one meter long before the change of definition. Would this stick still measure to be one meter long after you changed the definition of the meter?
Yeah I know quite offtopic but still... :P

 

[rbj]

Yeah that sounds understandable enough. I know most of our units are just arbitrarily chosen

not all of our units. just the base units. derived units (like the Newton for force) are derived from the base units, not arbitrarily chosen.

and have been changed many times*, I just didn't figure out this allowed us to make important constants equal...

*It's correct that the meter is now (since the speed of light has been defined exactly) defined to be the length that light travels in 1/c seconds, right?

that's correct since 1983.

And the second is defined as some large number of some transitions in the hydrogen atom, right?

not hydtrogen, but another element. you might want to check out the NIST site to get the definitions, history, all that poop. also check out the Wikipedia entries on those units and the SI system.

Hasn't this essentially changed the length of one meter?

when they redefined the meter, they did it at a time that, with the current definition of the meter, the resulting defined speed of light was exactly equal to their best guess at the measure speed of light previously. so meter didn't change in value at that time. but what did change were its future values compared to if they hadn't changed the definition.

Let's say you have a stick one meter long before the change of definition. Would this stick still measure to be one meter long after you changed the definition of the meter?

immediately after the definition changed, yes. 200 years after the definition changed? then no. not quite the same.