Law of Conservation of Energy and Symmetries of nature

Conservation laws are among the most fundamental concepts of physics. They enable us to solve mathematical and physics equations which are otherwise painfully difficult or even impossible to solve.

 yet, they are wrong. Atleast they're only true upto certian extent.

These laws are the consequences of the basic symmetries of the nature or more fundamental principle, Noether Theorem. (Read as NOT-ER)

Conservation laws are cheat codes of physics for solving complex and impossible math equations. They emerge from the simple and basic tools of reality. The connection between conservation laws and basic natural symmetries are encapsulated under noether's theorem.

The seeming paradox that led to the discovery of this noether's theorem is Einstein's general relativity in 1915. This paper had opened as many questions as it answered. 

Courtesy: Wikipedia
Einstein's General Relativity Paper published in 1915

Among them was the energy conservation. The Einstein's universe breaks the laws of energy conservation and predicts that the energy is not always conserved. 

The very common and simple example in this regard can be taken from cosmological red shift. In an expanding universe, as the distance increases, the energy of photons emitted from the stars loose energy continuously while travelling. Then where does this lost energy go?

No one knows!

In 1915, the expansion of universe hasn't yet been discovered. But, the failure of the law of energy conservation from the math of general relativity was discovered.

Two of the greatest mathematicians of the era, Felix Klein and David Hilbert, saught the help of Emmy Noether to understand what may be called a paradox. 

She discovered why the law of energy conservation broke down in general relativity.

 The law was not fundamental after all. She found that all those conservation laws are governed by more basic principles relations of universe. The basic symmetries of the system. These principle are what we call as noether's theorem.

A simple definition of Noether's theorem is as follows..

" For every continuous symmetry of the universe there exists a conserved quantity."

Lets unpack this now..

We say a face is in symmetry if it looks the same in a mirror reflection along a axis. A snow flake is symmetrical in 6 distinct angles. Playing cards are symmetrical in 180⁰ rotation.

                                        Courtesy: Space Times

These are called discrete symmetries. ie., they're symmetrical upto only for certain degrees of rotation. 

But, Noether's theorem applies only for continuous symmetries. 

We term something as continuously symmetric if it stays the same for any size shift in a given co-ordinate system.

For example, a road in between nowhere is continuously symmetric in spatial translation along the axis of the road.

Courtesy: Space Times
A road at nowhere having symmetry along axis of the road

A sphere is symmetric under rotational translations.

Courtesy: Space Times
A perfect sphere having rotational translation symmetries

In both cases, the environment stays the same but shifts the co-ordinates. In Noether's theorem, when we say environment stays the same we mean that the equations that give the motion of the object in the system remain the same.

Consider a flat road. The gravitational force along the road is constant and we have a spatial continuous symmetry along the road.

Courtesy: Space times
A road with gravity field indicated

Now assume a car is moving along this road. Then the noether's theorem predicts another conserved quantity called momentum, in this case linear momentum.

Then what about on a hilly road?

Courtesy: Space Times
A hilly road with gravity field indiacated

The system doesn't appear to be in symmetry here due to the variations in the gravitational field. This is because the fact that the gravitational pull direction always changes in this case along the road. But, on the other hand the gravitational pull doesn't change in the time frame IE., the gravitational doesn't change with time. Thus, the system has symmetry with reference to time, thus there is time translation symmetry. Noether's theorem reveals that this time translation symmetry gives us energy conservation.

Another classic example, the spherically symmetric gravitational field experienced by the satellites orbiting the earth, then the noether's theorem predicts another conserved quantity called, Angular Momentum.

Courtesy: The Science Explorer
A satellite under gravity field lines

By revealing the underlying source of conservation laws, the noether's theorem plainly explains when and why they're broken. That involves the apparent breaking of energy conservation in general relativity.

Einstein's description of gravity reveals the dimensions of space and time to be dynamic, changeable. If the nature of the space and time is changeable, the continuous time symmetry is broken.

Courtesy: vn.123rf.com
Einstein's Proposed Gravity Field

 That's the case with expanding universe. That's the same case in the cosmological red shift. Energy can be lost in some cases and it can be generated from nowhere in case of dark energies.

Whoa.,!

Wait a minute. Let's dive away from the topic a little.

From this explanation that energy can be generated from nowhere. Does that mean the concept of god creating the universe is real? can this nowhere be god ? YES? NO?

at least, we found a lead though!

Continuing,

                        The law of conservation of energy is fundamentally Newtonian mechanics in which time and space are unvarying and eternal. Thus, they're not valid in the real world. In Einstein's universe the law of conservation exists only as a special case. It is applicable where we can approximate space as unchanging with time.

Despite its profound implications, the math behind noether's theorem is surprisingly straight forward. It falls like magic from an other dimension of universe, the principle of least action.

This law states that the system in a universe chooses a path that minimizes the change it had caused in the first place. This is a abstract quantity that measures the dimensions of the system. 

If you remember, this is same as the Fermat's principle we've studied in high school physics. A light ray reflected from a surface always takes up a path that reduces the change caused by the action.

The principle of least action can be used to derive classical motion equations to quantum path integral functions. In other words this principle is Axiomatic. Its a founding assumption behind these derivations and also behind the derivation of noether's theorem.

Noether's theorem allows us to figure out the true conserved quantities for any system that is evolving according to the principle of least action, as long as we can identify that system's symmetries.

Thus, we've justified the law of conservation of energy.

Interestingly, my investigation and literature survey of this article has imposed some seriously un intended results. The scientific explanation for god and other supernatural powers might be possible in future. Science and god are not separate after all.