Gravitational Anomaly: (asked by a student for very simple explanation)
Basically it means the new laws of physics known as Quantum Mechanics invalidates the sanctity of nature’s principles or laws (that is QM brings exceptions to the validity of the physical laws of nature itself)
Let us discuss this in simpler ideas from the basics only.
Remember the most basic physics, that of principle of conservation of energy and the principle of conservation of (linear) momentum.
In the more rigorous formalism of physics these two principles emanate (that is derivable) from two ideas of symmetry. In-fact every conservation principle of physics are manifestations of a corresponding principle of symmetry and vice-a-versa (also every symmetry must correspond to a conservation law). This general idea of connection between conservation laws and symmetry is collectively known as Noether’s theorem and is a central underpinning in all of today’s conceptual physics.
What is symmetry? Let’s say you rotate your wheel-chair and yet the environment surrounding you in the room did not change, you see the exact same thing as before you revolved, then the fact that the environment is same over the rotation is known as rotational symmetry. Hence when some particular parameter is not changed while another parameter is transformed or changed, the invariance or lack of variation is known as a symmetry under that particular variance or transformation.
Another example is translation. You are going through a bridge. You see the exact same railing that you saw a little before. The translation transformation did not change the structure of the railing. Same thing happens within a crystal, it shows both rotational and translational kinds of symmetry.
Then in accordance with Noether’s theorem it follows that principle of linear momentum conservation follows from symmetry of translation in space. The system has a property known as linear momentum which does not change when the position coordinates describing the system are changed in parallel with themselves (known as translation) Principle of conservation of energy follows from symmetry of translation in time. The system has some property known as energy which does not change when time passes. Similarly principle of angular momentum follows from the symmetry or invariance (which means no change) of the system wrt angular coordinate transformation (known as rotation). Every other conservation principle (eg charge conservation) can be associated with its own unique way of keeping the system symmetric under some transformation. One thing changes, the other does not. The thing or quantity that changes is known as transformation and the one that does not is known as symmetry.
Now lets us go a little deeper. What is space, what is time and what’s classical mechanics?
Imagine a point particle, eg a small iron ball which can be represented by a geometric point so that its extension can be of no consequence to our discussion of its motion and related behavior. In more complicated situations the extensions can be considered but for our conceptual purpose we are not interested.
You can also imagine a molecule or atom which occupies certain position. When there are more than one such particle they need to be organized into a system which necessitates discreetness of the structure. In other words there is something that is there between the two different position occupied by two particles. This defines space. The organization of the particles continues to change, the distance between the particles change. Thus we need another parameter called time to describe motion that is to describe or define the change in organization.
But space and time are not ontological. That is they do not have any fundamental reality associated with them. They are merely construed from the necessity of the conscious observers (such as human beings). This defines Classical Mechanics.
All the principles and methods and ideas up-to and including the work of Einstein was mostly classical in nature. This means the two important works of physics known as special theory of relativity and general theory of relativity are classical in their scope although they are mathematically and conceptually very different from the kind of work that was done by Newton or Galileo.
So both Einsteinian and Newtonian mechanics (study and laws of motion) are same in the sense that space and time are not fundamental reality. That is why they are both included as classical mechanics. Special relativity is a new set of laws different from Newtonian mechanics which becomes relevant when anything moves fast enough in the limit of speed of light. This is possible only for particles with very small mass. But particles with very small mass (hence small size) are governed by new strange and crazy laws, known as Quantum Mechanics. (That means motion of a fundamentally small unit of matter).
Before Einstein space and time were considered as separate entities but in Einstein’s special theory of relativity emerged quantitative relations that placed space and time on equal footing. Also special theory showed that electric and magnetic field are both manifestation of the same field depending on how fast one is moving and in what direction. Einstein’s another marvelous work was general theory of relativity. This work showed that acceleration of an object is caused by a force which is not any different from a gravitational force. Which is why one would feel weightless in a satellite moving around earth. The acceleration is radially towards center of earth, which is equivalently a gravitational force radially outwards. Thus the acceleration cancels the effect of gravity. This is known as principle of equivalence of general theory.
But since space and time while equivalent are not fundamental reality they are merely a tool in the minds of the observers. Thus they must have no role in the physical laws of nature. When this is applied to all the ideas of special or general theory of relativity which are merely extension of Newtonian mechanics, the resulting mathematical ideas are known as general covariance. That is the space and time coordinate should appear in such a way in the laws of nature that when we go into another frame of reference the physical results won’t change.
This is greatly validated by everything we know experimentally or theoretically. But the new rules of mechanics known as Quantum Mechanics has many non-intuitive yet experimentally validated results which evinces its difference from classical mechanics. The difference is space and time could be a result of consciousness. But consciousness itself could be intertwined with reality. In other words we simply can’t disregard the reality of space and time. That is why Quantum Mechanics is non-classical (or modern) physics. Here space and time are indeterminate to arbitrary degrees of precision according to Heisenberg’s principle. This affects reality in a fundamental way.
Thus the calculations of Quantum Mechanics which are far more advanced than the basic maths of Einstein’s work for example introduced exceptions to the principle of general covariance. Remember that general covariance is what is central to the validity of gravitational interactions. It says that changing coordinates (such as space and time) should not change the laws of nature or physics. But Quantum Mechanics throws exceptions to the validity of this necessary principle. This is the reason why we do not have a completely successful theory of physics (known as Grand Unified Theory, dreamed by Einstein himself) till today. This exception to a natural idea known as General covariance is known as gravitational anomaly. This needs to be solved if we are to make progress in fundamental physics.