# Relation between energy and time

I would like help in understanding something that has been causing me a lot of trouble recently: Why is energy always related to time in physics?

Examples include the 4-momentum, the energy-time uncertainty principle and Hamiltonian mechanics. Specifically, how is the Hamiltonian the generator of time translation?

I understand where it comes from mathematically but I don't have the physical intuition behind it.

When it comes to translations and rotations, I think I have a good grasp as why it is like that. For example, I understand why a translational invariance implies a conservation of momentum, etc. But when it comes to energy and time, I don't have a clue of why it is like that.

Could someone please enlighten me on the relation between energy and time?

• Possibly useful: en.wikipedia.org/wiki/Conjugate_variables and the notion of “action” Commented Jul 14 at 23:54
• Are you asking what time is? Time is that which the clocks show. What is a clock? It's a local energy reservoir that releases this energy evenly towards infinity. As you can see, a Hamiltonian systems approach to time is probably not quite enough to explain "where time comes from" because time is intricately related to energy flow in space. Commented Jul 14 at 23:55
• Are you familiar with Noether's theorem? Is your physical intuition ok with conservation of linear momentum being equivalent to spatial translation symmetry, and conservation of angular momentum being equivalent to spatial rotation symmetry? Commented Jul 14 at 23:58
• the reason we have Energy, Momentum, Angular Momentum is because there is symmetry under time translations, position translations, rotations.
– JEB
Commented Jul 15 at 5:54
• Ok. You should clarify that in your question, so that answerers don't feel obliged to explain Noether's theorem from square one, and can focus specifically on the time-energy relationship, and how it ties in with the other relationships that I mentioned previously. Commented Jul 15 at 18:16

There are at least three occasions where the notions of Energy and Time show up together in classical and modern physics.

1. Probably the most elementary situation is related to the fact that the product of [Energy] and [Time] has the dimension of [Action], and there exists a fundamental physical constant with the dimensions of [Action]: the Planck constant. This mere fact gives rise to a number of interesting occasions where the two notions meet, first of all the well known Heisenberg inequality.

2. Another crucial fact is that the law of conservation of energy is a consequence of invariance under time displacements. This implication takes place in many formulations of dynamical laws of classical, special and general relativistic, quantum-relativistic physics. (It is the content of the variaous formulations of Noether's theorem for the considered case).

3. The third occasion where the two notions meet is much more advanced. It concerns a "curious" fact appearing in, once again, several formulations of physical dynamics in the statistical version. There is a subtle relation between time evolution and the description of thermodynamical equilibrium states. There time and inverse-tempertature $$\beta=1/(k_BT)$$ are interconnected under a sort of "Wick rotation". Here $$\beta$$ is coupled to energy, in the Boltzman factor $$e^{-\beta E}$$. In more sophisticated formulations, this interplay takes place in the so-called KMS relation. This sort of connection arises both in classical and in quantum situations, even in QFT and in the quantum theory of Black Holes (Hawking radiation).

Roughly speaking, in quantum theory, when passing from the real time to the imaginary time the quantum time evolutor $$e^{-itH}$$ becomes the operatorial representation of a mixed state at finite temperature $$e^{-\beta H}$$.

• I was going to bring up the second law of thermo. Commented Jul 15 at 18:46
• Is there any layman-accessible explanation of why time invariance forces energy conservation? I got gauge invariance very quickly: if we cannot measure absolute voltage, but the potential energy of an electron is proportional to the voltage, we cannot create or destroy electrons (at least not without converting to some other particle with the same potential energy) because the amount of energy used to create the electron would betray the absolute voltage. Is there any explanation of energy conservation that is so simple? Commented Jul 15 at 20:43
• I do not know, it is a fundamental case of all Noether like theorems (in classical mechanics this case is also known as Jacobi's theorem). I think there are some attempts in PSE to elaborate an elementary explanation of the relation symmetries - conservation rules. I never tried it, but I sugget you to look around in PSE for these attempts. Commented Jul 15 at 20:46
• @StackExchangeSupportsIsrael "Is there any layman-accessible explanation of why time invariance forces energy conservation?" There's no layman-accessible explanation that does not involve equations, since you need equations to define what you mean by "energy." And you usually also want to use equations to define what you mean by the "system," e.g., in terms of a Lagrangrian or in terms of external forces.
– hft
Commented Jul 15 at 22:12

Why is energy always related to time in physics.

I don't think it is helpful to describe energy as "always related to time" in physics.

That being said, there certainly are a number of interesting relationships involving energy and time, such as $$\frac{\partial S}{\partial t_f} = -E(t_f)\;,\tag{1}$$ and $$\frac{\partial S}{\partial t_i}=E(t_i)\;,\tag{2}$$ where $$S$$ is the classical action.

I won't prove the equalities in Eq. (1) and Eq. (2), but they can be motivated by considering the free classical action: $$S_0 = \int_{t_i}^{t_f}dt\frac{1}{2}m {\dot x(t)}^2 = \frac{m}{2}\frac{(x_f-x_i)^2}{t_f-t_i}\;,$$ for which Eq. (1) and (2) clearly hold.

When it comes to translations and rotations, I think I have a good grasp as why it is like that. For example, I understand why a translational invariance implies a conservation of momentum etc...

Translational invariance in space implies conservation of momentum. Translational invariance in time implies conservation of energy.

But when it comes to energy and time I don't have a clue of why it is like that.

If your Lagrangian does not depend explicitly on time then we have $$\frac{dH}{dt} = -\frac{\partial L}{\partial t} = 0\;,$$ so energy is conserved. (I will not prove the first equality above since it is well known.)

This is one formulation of time invariance (Lagrangian doesn't explicitly depend on time) implying that energy is conserved.

But probably is it better to think in terms of the action, where the invariance in time more clearly leads to the conservation of energy. (See below.)

Another good way to think about how time translation invariance leads to energy conservation is in terms of the action: $$S(t_f, t_i) \equiv \int_{t_i}^{t_f}L(x(t),\dot x(t)) dt\;.$$

Translational invariance of the action with time means that $$S(t_f+\delta t, t_i+\delta t) = S(t_f, t_i)\;,\tag{3}$$ which in turn implies that $$\frac{\partial S}{\partial t_f}+\frac{\partial S}{\partial t_i}=0$$ $$=-E(t_f)+E(t_i)\;.\tag{4}$$

Or, rearranging the terms in Eq. (4), we have: $$E(t_f) = E(t_i)\;,\tag{5}$$

In other words, time-translation invariance (Eq. (3)) implies that energy is conserved (Eq. (5)).

• Thank you for your answer. I understand your last paragraph mathematically it makes sense to me. However my real question was how can I understand this physically without the use of math. Maybe it is dumb of me to try thinking without the use of maths and maybe I should change of paradigm. Commented Jul 15 at 20:05
• @Lucas You can't really understand it without math, since the mathematical definition is usually needed from the begining. For example, within non-relativistic classical mechanics, the kinetic energy is defined as $T=\sum_i \frac{1}{2}m_i \left(\dot x_i^2+\dot y_i^2 + \dot z_i^2\right)$. For example, conservative potential energies are defined via $\vec F = -\vec\nabla U$. For example, the Hamiltonian energy is defined as $H=L-\frac{\partial L}{\partial \vec v}\cdot \vec v$.
– hft
Commented Jul 15 at 21:50
• Yes, I think that I should change my way of thinking about it. And maybe consider math into my understanding of the problem Commented Jul 15 at 21:59

The way I like to look at it (and which may or may not give you the same amount of intuition as it does to me) is as such:

• Momentum is what gives rise to changes in position. Clasically, a body having momentum will change its position. Momentum is also known to be the generator of translations (i.e. changes in position), e.g. in QM.

• Angular momentum gives rise to changes in angle. A rotating object will rotate, trivially. Angular momentum is the generator of rotations (changes in angle).

• Similarly, I like thinking of energy as a generator of time evolutions, i.e. a system with non-zero energy must evolve (non-trivially) in time. I admit it paints less of a clear-cut picture than the other two cases, but it seems to make sense: the energy content of a system, if kinetic, will cause constituent particles to move and the system to change with time (think ideal gas in a box), even though the system may remain the same macroscopically. If the energy of the system is potential, it may cause the system to come into motion (think gravitational freefall from rest).

Also relating energy and time is the quantum mechanical Ehrenfest Theorem, which states that the time evolution of (the expectation value of) an operator is determined by the system's Hamiltonian.

Of course, these pairs of quantities I mention are very closely linked mathematically as @Valter Moretti's answer alludes to. Dimensions of the product, generators/conserved charges through Noether's theorem and Wick rotations are beautiful mathematical ways to think of the relations, and are the ways that prove to be useful in calculations at the end of the day.

I understand the question as the OP asking of an intuitive way to think about the relation between energy and time. Therefore the point I would like to make is that energy isn't really a thing. So for example potential energy and the oscillating magnetic field of a photon have nothing in common. It is just a measure of the ability of a system to cause an action. This ability can be converted from one system to another and quantified. So the idea of energy is really just an abstract concept.

So every physical formula having E one one side will have an expression with t on the other side, because it describes the change that happens when a system is converted from one state to another within a specific amount of time. The Hamiltonian describes exactly this evolving system without the explicit use of the energy E.

A good example for that is potential energy which is described by m * g * h. It is only detectable by having an object fall and accelerate in relation to t². t² is required to describe Earth's acceleration g which is expressed in meters per second squared (m/s²).

The 4-momentum is actually a good example to show this. Depending on the relativistic conditions time may pass differently between observers resulting in one observer detecting a different momentum than another, despite observing the same system. Even the observed change of state appears different which would imply a difference in energy. But although an object moving at relativistic speed would appear to move slower the observed mass would increase resulting in the same observed momentum and energy. So time is even influencing mass which isn't too surprising, since energy and mass are equivalent.

• I'm not the downvoter, but the second paragraph needs clarification. The expression mgh does not have a variable "t". Also absolute gravitational potential is not detectable. Commented Jul 17 at 1:43