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How accurate are physics laws? For example, for newtons' first law $F=ma$, if we can get a measurement of both force, mass and acceleration with a percentage of uncertainly close to $1\times 10^{-9}\%$, will the formula match the value we determined? If not, how many percentages of error could we take and still believe the law still hold?

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  • $\begingroup$ Might there be a difference between the laws of physics and the results you or I or anyone else calculates? $\endgroup$ Apr 1 at 23:45
  • $\begingroup$ @RobbieGoodwin Yes: firstly, besides the errors in calculations, they may use different approximations, or take into account different effects relevant to experimental situation; secondly, what one calculates and what one measures are rarely the same thing. $\endgroup$ Apr 2 at 8:05
  • $\begingroup$ @Vadim Thanks and that wasn't quite my point. What you said was true but it applied only to the experiments, not the laws. Whether you concentrate on calculation or measurement still, I sincerely hope the Question had nothing to do with differences of that kind. For example, with 𝐹=𝑚𝑎, if we work with accurate measurements of the initial conditions, why might we not expect accurate results? Why might 1×10−9% matter in terms of the laws? $\endgroup$ Apr 5 at 20:12
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Accuracy can mean different things. While the question asks about the statistical accuracy, what immediately comes when talking about the Newton's laws is that they are non-relativistic, i.e., they are valid up to small corrections of order $v/c$.

Physics laws are based on empirical observations, the symmetries of the universe, and approximations appropriate for a given situation.

Symmetries
For example, we have reasons to think that conservation of momentum or energy are exact laws, since they follow from the symmetry of space in respect to translations in space and time (Noether's theorem). Testing these laws in practice will necessarily result in statistical errors, but improving the precision of measurement is unlikely to uncover any discrepancies.

Approximations
Newton's laws are valid only in non-relativistic limit. Thus, they will hold only up to small corrections of order $v/c$ where $v$ is the speed of the object and $c$ is the speed of light. If our relative statistical precision (in measuring the force, acceleration, etc.) is of order $v/c$, we will observe deviations.

Empirical observations
Laws of thermodynamics are a good example of the laws that were deduced phenomenologycally, as a result of many observations. Yet, statistical physics shows that they hold up to very high precision ($\sim 1/N\sqrt{N_A}$, where $N_A$ is the Avodagro constant). If the precision could be so high or when dealing with systems where the number of particles is not small, we will observe deviations from these laws.

Remark
I recommend the answer by @AdamLatosiński, which is technically probably more correct than mine. What I tried to explain in my answer is how the laws of physics are different from, e.g., the biological laws (since the subject was recently debated on this site) - the latter are generalizations of many statistical observations, but not grounded in reasoning about fundamental properties of the universe. They are therefore statistical laws, which are bound to be non-exact. Indeed, even the so-called Central dogma of molecular biology ($DNA\rightarrow RNA \rightarrow Protein$) is broken by some viruses, performing reverse transcription ($RNA\rightarrow DNA$.)

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  • $\begingroup$ I believe it's the difference between the precision and accuracy. The measurement may be imprecise, the law may be unaccurate. $\endgroup$ Mar 31 at 7:23
  • $\begingroup$ @AdamLatosiński yes, indeed - my thinking was in a different direction, butw hat you say is correct. $\endgroup$ Mar 31 at 7:25
  • $\begingroup$ What do you mean by the term "statistical accuracy". Accuracy is stated going either way (measured value compared to law or law compared to measured value) by using only one measured value and only one "truth" value. We need not invoke statistics. Also, what other "different things" can accuracy mean that are equally valid? Isn't accuracy in this case simply "Is (law X) different from (observation Y) to this degree of confidence? $\endgroup$ Apr 1 at 2:01
  • $\begingroup$ @JeffreyJWeimer Accuracy here is meant as a non-technical term. The accuracy that you are talking about is what I called statistical accuracy, as it refers to measurement. However the "truth" itself is inaccurate (as in phrase accuracy of physics laws). Note that confidence intervals is actually a very statistically involved concept from hypothesis testing (see my other recent answer here physics.stackexchange.com/a/626076/247642) $\endgroup$ Apr 1 at 6:56
  • $\begingroup$ I will follow up separately. $\endgroup$ Apr 1 at 13:32
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There are errors that come from measuring the quantities and errors that come from the inaccuracy of the laws themselves.

If we know only approximate values of parameters in the equation, then we can calculate how precise the result it using formulas for the propagation of uncertainty. For example, for the formula $F=ma$ we have $$ \frac{\sigma_F}{F} = \sqrt{\left(\frac{\sigma_m}{m}\right)^2+\left(\frac{\sigma_a}{a}\right)^2}$$ where $\frac{\sigma_m}{m}$ is the precision of the measurement of mass and $\frac{\sigma_a}{a}$ the precision of the measurement of accelerations. When you measure all three quantities, you can then check whether $\frac{F}{ma} = 1$ with the precision obtained by the appropraite formula, specifically you calculate $$ \sigma^2 = \left(\frac{F}{ma}\right)^2 \left(\left(\frac{\sigma_m}{m}\right)^2 +\left(\frac{\sigma_a}{a}\right)^2 +\left(\frac{\sigma_F}{F} \right)^2 \right) $$ and, given measured values of $m$, $a$, $F$ you calculate $\left|\frac{F}{ma} - 1\right|$ and compare with $\sigma$. If it's of order of $\sigma$ or lower, it's generally considered within expectation: these kind of error may be just a result of imprecise measurement. If it's around $3\sigma$, then it may be a reason to check whether your measurement was done properly, but it can still happen from time to time. A deviation much bigger than $\sigma$ means that either your measurement is done wrong or the law is unaccurate.

Speaking of accuracy: regardless of the precision of measurement the laws themselves may be more or less accurate depending on the situation. For example, the Newton's law needs corrections that get bigger the closer the velocity is to the speed of light, and a corrected formula would be $$ F = \frac{ma}{\left(1-\frac{v^2}{c^2}\right)^\frac32} $$ For $v=1 \text{ km/s}$ we have $$ \frac{1}{\left(1-\frac{v^2}{c^2}\right)^\frac32} \approx 1+1.67 \cdot 10^{-11}$$ while for $v=200 000 \text{ km/s}$ we have $$ \frac{1}{\left(1-\frac{v^2}{c^2}\right)^\frac32} \approx 2.42$$ So the accuracy is good for low velocities (if your measurement precision is of order $10^{-9}$ you won't see any unaccuracy) and weak for high velocities (easy to notice even with low measurement precision).

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    $\begingroup$ Isn't Newton's second law, in relativistic form, $\mathbf{F}=\frac{d}{dt}(\gamma m\mathbf{v})$, and not $\mathbf{F}=\gamma m\mathbf{a}$? $\endgroup$ Mar 31 at 17:57
  • $\begingroup$ Newton formulated his law as force equal to rate of change of momentum (called by him "impetus" I think), not mass times acceleration; he was quite clear and explicit on this point. Relativity changes the relationship between momentum and velocity rather than the one between force and rate of change of momentum. But this does not affect the main thrust of this answer. $\endgroup$ Mar 31 at 20:40
  • $\begingroup$ @JohnDumancic You're correct. I've corrected the answer, taking into the account that $$ \frac{d}{dt}(\gamma v) = \gamma^3 \frac{dv}{dt}$$ $\endgroup$ Apr 1 at 1:31
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    $\begingroup$ Perhaps rather than saying that laws may be more or less accurate "depending on the situation" you might better say "depending on the the truth in the ASSUMPTIONS". By kinetic theory, nothing is an ideal gas. Yet, we engineer the behavior in real processes by the ideal gas law more often than we care to believe. It is not the situation that makes it valid, it is the high degree of confidence in the assumptions that allows us to trust it as valid. $\endgroup$ Apr 1 at 2:05
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    $\begingroup$ @JacktheRanger I've assumed that the variables are independent, but if they weren't there would be an additional term. $\endgroup$ Apr 2 at 9:09
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"Physical Laws" is a misleading expression. Physical theories and models expressed as mathematics are what we have. That does not mean the universe works that way "under the hood", it means those rules are good models to predict what will happen in given circumstances.

How Accurate a given theory is depends on three things :

  • How accurate you need it to be. Why expend computational effort on a complex theory when a simpler one is "good enough" for practical purposes ?
  • How accurately you can make measurements to base your theory/model on. If I cannot make measurements accurately enough to check the predictions of my theory, how accurate it is is impossible to say.
  • How practical is it to perform the required mathematics to make predictions. The vast majority of practical mathematics in physics is approximate as the full blown theory is not possible to work out explicitly.

Here's a thing to remember. The number $\pi$ turns up all over the place in mathematics (and hence in physics). Likewise the number $e$. But these numbers cannot be calculated exactly - no matter how you do it you need an infinite number of digits to be accurate. So no theory, in practice, can get beyond what we can practically calculate. Taking the $sine$ or $cosine$ of a value is likewise something you will not in general be able to do with unlimited accuracy. Now these common functions and values have very, very precise methods to use in practice, but there are many more common functions in physics that we just have to truncate to make it practical to work with. Something as trivial sounding as e.g. the length of an ellipse's perimeter turns out to be impossible to calculate exactly in numerical terms. So mathematics is a practical limitation in many ways.

A very common approach used by physicists is to use a linear approximation to a non-linear model simply to facilitate using it in other models. So we can say that $V=IR$ (relating voltage, resistance and current in a resistor), but we know it's not really that simple. For some purposes that's as much accuracy as you need, for others you need additional terms and it becomes extremely complex. There are multiple models of gas laws which are, in different end use cases, better or worse than each other. It is surprisingly common to see the simplest one ($PV=nRT$) turn up places you know they are not perfect for e.g. white dwarfs. This is because we don't always need accuracy, we need a qualitative idea of what happens and numbers that are in the ballpark are good enough for that purpose.

If not, how many percentages of error could we take and still believe the law still hold?

So after all that you will see that the law "holds" as long as it is accurate "enough" for the purpose we need.

We don't completely stop using e.g. Newton's Laws of motion just because we have relativistic theories that are better in absolute terms. Newton works fine for so many purposes that we stick with it - There is nothing preventing a physicist going their entire career and never needing e.g. relativity (after they learn the basics for an exam).

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    $\begingroup$ The exact value of 𝜋 is 𝜋. It doesn't have an exact decimal representation, but most exact numbers don't. You may reason with mathematical exactitude using 𝜋, as with any other exact number. For example, sin(𝜋) is exactly zero. But of course, all such mathematical reasoning refers to products of the human imagination: 𝜋 represents properties of such objects. Often, such objects can effectively model physical objects, but one must not confuse the model with reality. $\endgroup$
    – John Doty
    Mar 31 at 20:54
  • $\begingroup$ The answer seems to carry a fallacy. The number of significant digits in a number is not a metric of its accuracy. It is a measure of its precision. $\endgroup$ Apr 1 at 2:14
  • $\begingroup$ @JeffreyJWeimer The OP asked about the accuracy of theory and as theory is pure mathematics whereas actually calculating values is fundamentally limited, the precision you are prepared to apply is relevant. Note it's not just numbers that are the issue - working with many mathematical functions is abstract and it is not always possible to treat them precisely within the limits of existing mathematics. I can write down an equation describing a system exactly within a given theory, but that does not mean I can solve that equation. Even numeric solutions can have high errors. $\endgroup$
    – StephenG
    Apr 1 at 13:47
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The laws of physics aren't expected to be infinitely accurate. For example, it's expected that at Plancks length, $l_P$, which is approximately equal to $1.6 × 10^{-35}~\mathrm{m}$, and which is related to Plancks constant, is the scale where the continuum description of length, area and volume breaks down.

This is one reason why in Loop Quantum Gravity there are area and volume operators - and probably length operators, although I haven't checked - whose spectrum gives the minimal distance probeable and also why in Causal Set Theory, the discrete spacetime structure is taken as axiomatic. This also happens to be another, radical attempt at Quantum Gravity that doesn't try to quantise gravity - the bottom up approach - but builds it instead ab initio - the top-down approach.

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  • $\begingroup$ Why should this happen at the Planck length? Why not e.g. at energy scales of one Planck energy? $\endgroup$ Mar 31 at 20:27
  • $\begingroup$ @Jannik Pitt: Because I was using the Planck length as an example and besides the OP was asking about Newtons Laws and these rely on spatial measurements. $\endgroup$ Mar 31 at 20:34
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    $\begingroup$ The point was that there is nothing special about the Planck units. One Planck energy is about $10^9 J$, one wouldn't expect any interesting physics happening there... $\endgroup$ Mar 31 at 20:48
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The answers given and accepted provide a well-structured set of responses.

One confusion still seems to remain. A physical law is a concisely-constructed statement about the behavior of a system or process. It is constructed to explain empirical observations in nature, and it has never been found to have been violated through the course of all past observations. We should not in this sense ask whether a law is or is not accurate. To the extent that a statement becomes accepted by calling it a law, it is accepted to be valid. We don't question its accuracy further as as statement unto itself.

Let's consider an example for a better framework on the potential confusion. The ideal gas law encompasses a statement about the observed behavior of gases under certain conditions. Real gases deviate from this law as we take them to lower temperature and higher pressure. We could propose to say that the law is inaccurate as we move to lower temperature and higher pressure (real gas behavior). I contend that we will instead be better when we say that the conditions present when the observations are made violate (go beyond) the boundaries set on the conditions required for the ideal gas law to be valid or applied to the system at hand.

It is not that the law is inaccurate as a way to represent the observation. It is that the system conditions are invalid to allow us to apply the law.

Low temperature and high pressure can have nebulous meanings as boundaries. Let's use kinetic theory or the universality framework of gas laws to express the nebulous boundaries in more meaningful ways. For any given temperature and pressure, applying the ideal gas law to predict or calculate the behavior in a system will be less valid (not less accurate) for molecules that are larger in size, have greater interaction energies (secondary bonds), or have greater degrees of inelastic collisions (viscosity). Alternatively, at the same temperature and pressure, applying the ideal gas law to predict or calculate the behavior in a system is less valid (not less accurate) for a gas at lower values of reduced temperature or reduced pressure.

Perhaps also an inverse example will help. Suppose that we make a measurement and the value does not agree with predictions from a certain law. We should not say that the law is inaccurate. We should categorically say that the measurement is not accurate against the given law because the law is invalid at the conditions present during the measurement.

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I'd like to offer a simple analogy as an answer.

How accurate is a photograph, of some object, when its printed or on a monitor?

Well, its pretty accurate. That 4K monitor sure seems to show the photo of that street exactly as it looked. Except, of you look really closely, doesn't it have pixels/dots? That street scene wasn't pixelated, or was it?

Okay, ignore pixels, the colour looks right. Top notch calibrated camera, monitor and software, it should be right. Except that monitor isn't really displaying orange photons as orange, its creating the illusion of them by fooling the eye with 3 primary colours in a tiny area.

Well, suppose we let that go, and look at purely red, green and blue objects - surely those are okay? Except, that monitor only displays their brightness to 8 or 10 bit precision, because we cant see more with the eye, but photons aren't limited to those levels. So even that is wrong, its just good enough to not see it with the naked eye.

So those photos as displayed, are not the right smoothness, or colours, or levels. So why do we think it's a good likeness at all? Because its so close, we can live with the approximations between our monitor's version of that scene, and the reality of it. We can use it the same way that Google Maps doesn't show some discarded beer can or wobbly roadside kerb, and it's still a really good guide for navigation.

But we know that if we probe deeply enough, we will probably find cases where our representation - the photo or map - isn't quite spot on. When that happens we don't declare the entire photo or map useless, we just acknowledge it always was and always will be, our best attempt to match what we understand reality to be, and with limits set by the task. For navigating to my workplace, I don't need the beer can or kerb on the map. For a crime scene I might need a highly calibrated full colour gamut photo.

And it's like that with physical laws too.

We don't know how the universe works. If we wake up tomorrow and fond it's different than we thought, that'll be reality (or delusion), and we have to deal with it. But we do find the universe so far seems to work in a way that matches very very tightly, certain formulae.

If we test and test and refine and refine those formulae, and they prove out very and over again, we might call those formulae, and the matters they represent, "Laws of Nature" or "Physical Laws".

But at best, they're just like our monitor. They are best treated as approximations of whatever reality may be, and how it seems to behave. The formulae can show us deep insights, they can predict things that we can go our and look for - and now and then they show us they are an incomplete set of formulae, because something happens in reality that isn't captured or doesn't match those formulae.

When that happens, we check reality again (ask other teams to check our findings), and then if we decide our formulae were not quite correct, we try and refine them and make better ones that keep all the strengths of our existing ones, and also match reality for the new observations as well.

And,because we expect this to happen, we constantly recheck our formulae against reality, to ever more precise and arduous tests. But we can never say "that's it", because any day, we could find a situation that our formulae don't get quite right.

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