THE MOST BEAUTIFUL IDEA IN PHYSICS: CONSERVATION LAWS? NO!!! NOETHER'S THEOREM

Ever wondered about why energy, momentum, etc. is conserved before applying them to solve any problem in physics. Well, if you have then you are in a treat. Conservation laws feel us fundamental as it gets and yet they are only right sometimes. To actually know what quantity is conserved at a given specific condition and time we require symmetries. Yes, it does sound a bit weird that how symmetries can influence the nature of mathematical quantities within a system to be conserved but they actually do.

This is shown by Emmy Noether in 1915 and published in 1918. It states that 'For every continuous symmetry in the universe, there exists a conserved quantity'. Now, what do we mean by a continuous symmetry? 

For Instance, a perfect butterfly is symmetric by a mirror reflection, as seen in the picture below.

Mirror Symmetry of a Butterfly

As well as the translation symmetries of a crystal lattice.

The symmetry of crystal lattice 

However, these are not continuous symmetry rather than a specific length of space. A continuous symmetry means the object stays the same for any size shift in the given coordinate, no matter how small. For example, a perfect sphere is continuous symmetric under rotational translations.

Continuous Rotational Symmetry of a Sphere 

Conservation of Energy: 

According to this idea, in a given system, energy is conserved, if and only if the system is continuously symmetric over time known as Time Translation symmetry. This means that no matter when you do the measurement, it gives you the same outcome. Thus, the conservation of energy works for rigid space-time and for short distances in space-time. But on the scale of the universe, space-time is not rigid, rather it is flexible as suggested by Einstein.

This tells us that conservation of energy is not a fundamental idea despite the fact that it feels very fundamental. Let's take an example of rigid space-time where space does not change with time. In this system, the energy is always conserved because no matter where in time, space looks exactly the same and the system is continuously symmetric over time. But we know that space-time is not rigid rather flexible, thanks to Albert Einstein. The space-time is, in fact, expanding in time. so if a light of specific wavelength starts to travel in space over a huge distance of several light-years its wavelength will expand as well, meaning the energy of the photon will be decreased over time. Where does that energy go? It is lost to the space-time fabric. and not conserved after all. This discovery is called the Cosmic Redshift.

An Illustration of Cosmic Redshift

If In Dynamic Space-time Energy is not Conserved then what is?
It turns out that we can figure it out by Noether's Theorem as well. It is known as Landau-Lifshitz Pseudotensor. The mathematics of this is very complicated and I am just an engineering student trying to learn physics so I do not have a good understanding of this tensor. However, If you want to learn more about this here is the link Landau-Lifshitz Pseudotensor.

The founding Assumption behind Noether's Theorem and Landau-Lifshitz Pseudotensor is The Principle of Least Action which I will be covering in the coming posts. So stay tuned for that.