A legit definition of Symmetry is that the "invariance under a specific group of transformation". Symmetry is more than a circle symmetric along its diameter, there should be no change in some transformation to out loud that the condition is symmetric. Old times of Aristotle and a group of philosophers measure Symmetry as a beauty factor in something. But the intuition was implicit only. Today we have laws, principles that govern the properties of a system including symmetry. It is more of a quantum thing at present but still, history talks about a lot of magnificent uses. The dynamic equations in Quantum mechanics are governed by Hamiltonian or Canonical formulation. So the system should be unchanged under this kind of formulation.
In relativity, the idea goes with the frame of reference or the coordinates in some frame to be specific. The transformation from one coordinate system to others. There exists coordinates transformation i.e Cylindrical, Spherical. This does not imply any effect on the process and the system in which the entire process is happening. It is a way to measure it in some different manner.
The general theory of relativity deals something more general than others it is known as 'General Covariance'. It is more or less the measurement of quantities in a different manner. It could be explained in the form of temperature. As we know that temperature can be measured easily by a device known as a thermometer. The function of the thermometer is to measure the heat, not the value of temperature because the method of defining us totally depends on us. We can change in whatever way we want to, such as we measure it degree Celsius, Fahrenheit, Kelvin. The values are different but all the different kinds of scales measure the same heat. 0 degree Celsius and 273K are the same things, so they are said to be symmetric.
C, P, T
The symmetries are aligned in a manner such that in QM there exist some ways to pronounce their nature. "C" corresponds to Charge conjugation. This goes with the invariance in the form of charge. If we are talking about a particle then we should know that there exists an antiparticle. Now, this antiparticle is the same in all the manner except the charge. So, we can say that this is a factor in which symmetry exists.
"P" corresponds to Parity. It is the only weird sounding among three and trusts it is the one with which you are familiar most. It is the symmetry that exists when you place something in front of a mirror. The image that reflects is a symmetric image of the object.
It can be observed experimentally in chemistry. There is a property in some compounds known as Chirality, which deals with the alignment of bonds. The only change in the alignment of those bonds changes the entire form of interaction with some other compounds.
"T" deals with time, a symmetry which we cannot simply observe in nature but it will never cease to exist. It is the symmetry of the flow of time forward and backward. In terms of thermodynamics, we can say that it represents the increasing and decreasing entropy. We know the entropy of the universe id increasing with "time". This basically corresponds to the inevitable "arrow of time"
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