What exactly is meant by infinity?

Question

1. What exactly is meant by infinity when I see it in a physics equation (always something wrong?)?

2. And in experiment how many orders of magnitude can be treated as infinity (say, if infinity is predicted by theory)? (I would like an example, since $10^{10} sec$ can be big comparing to a blink of eye, or small comparing to the age of universe)

Answer 1

The concept of infinity as used in calculus arises mathematically as an equivalence class of divergent series to compactify the reals (or complex numbers), so infinities are, even in a purely mathematical (calculus) setting, in some way approximations of large numbers.

There are rich theories of other concepts related to infinity not directly related to the concept of infinity occurring in physical equations, such as infinity of sets (which can be classified more finely by cardinal numbers), infinite series (which are of course at the heart of calculus), but these are usually not surfacing directly in physics (or only in purely mathematical formalism, such as infinite sums).

In physics infinity thus typically arises in two situations:

• As a mathematical simplification/approximation (such as assuming infinte mass of a nucleus when calculating the energy eigenstates of hydrogen). In this sense we use the fact, that the observables converge as our parameter goes to infinity, so to the required degree of accuracy we can approximate a very large parameter as infinity.

• As divergences in mathematical transformations such as perturbation theory. These types of infinities are typically non-physical and must be handled on a by-case basis, by some form of regularization/renormalization or physical argument.

If theory predicts infinity, the theory is most likely incomplete or the infinity does not have observable consequences (the singularity of a black hole which is causally disconnected from the rest of the space time by the event horizon) or is not itself the value of an observable (such as the poles of Greens functions, which while carrying information about observables, namely the excitation spectrum, are not observables themselves).

How many orders of magnitude can be considered consistent with infinity in an experiment depends on the required accuracy and on the specific model and cannot be discussed in a general fashion. A better way to test for consistency with the theory (instead of trying to measure the "infinite" value) is to measure the observable close to the pole (if this is possible) and test how well the measured scaling behaviour is in agreement with the theory.

Answer 2

The meaning of infinity in an equation is always contextual. One example is the Lorentz factor, which shows the strength of relativistic effects

$$ \gamma = \frac{1}{\sqrt{1-v^2/c^2}} $$

If we set the speed in the equation to the speed of light, then the Lorentz factor goes to infinity. This isn't a problem, because objects can only approach, but never reach the speed of light. This can have reasonable interpretations even if we allow the Lorentz factor to go to infinity. For example, it takes infinity energy to accelerate something to the speed of light. The time between two events in our reference frame from the perspective of a photon is infinite.

Another example is the input impedance of an antenna. A typical analysis of a dipole antenna predicts infinite input impedance at anti-resonant frequencies. In real antennas though, you will measure large but finite impedances, in the range of 500 Ohms. In this case, we recognize that the infinity came from a simplified model, and only expect a "large" impedance, rather than infinite (a typical impedance at resonance is around 70 Ohms, so 500 Ohms is considered large in this context).

Then there's the infamous problem of renormalization. I don't entirely understand it, so I can't explain in further detail, unfortunately.

Answer 3

Mathematical approximation of physical models is one reason.

Take for example the spring mass oscillator where damping, assumed to be negligible is entirely discounted. A linear systems model in terms of a Laplace transfer function relating displacement of the mass relative to applied force is

$$\frac{x(s)}{F(s)}=\frac{k}{s^2+\omega^2}$$

The roots of the characteristic equation (denominator in the equation) are a complex pair of imaginary 'poles' which represent a frequency at which the response of the system becomes infinite. Indeed by driving this model with a force $f(t)=\sin(\omega t)$ , $s^2 \to -\omega^2$ and one predicts an infinite displacement.

But no physical (real world) system has the capability of infinite displacement, nor the the ability to store infinite energy. Physical systems do have losses (damping) and non-linearity (saturation limits for example). And so the system model fails to predict what actually happens due to invalid assumptions and lack of fidelity.

Answer 4

always something wrong

No, its not wrong, but mathematically you cannot calculate with the infinity variable $\infty$. Instead of inputting the infinity variable on another variables place, you should try to figure out what limit/value the function will reach, if you did.

In texts you will often see something like

$f(x)$ goes towards $...$ when $x \rightarrow \infty$

(or maybe said less mathematically) and never

$f(\infty)$ has the value $...$