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According to Emmy Noether’s symmetry theorem, all conservation laws of nature are derived from fundamental symmetries.

Translational symmetry makes conserved linear momentum. Rotational symmetry makes conserved angular momentum. Temporal symmetry makes the conservation of energy possible. But she did not emphasize how these symmetries are viewed and found to be valid. In this discussion, these symmetries can be viewed using two basic points of view, locally and globally. It is also the purpose of this discussion to give some brief (but not exhausted) analyses how these three fundamental symmetries can be viewed locally as well as globally and it is hoped that some deeper differences between each perspective can be found without further rigorous discussions or the use of heavy math.

Translational symmetry – Newton’s 1st law of motion contains this symmetry giving the definition of a vector. Linear momentum is the product of mass and velocity but only velocity is a vector. Mass is a scalar. Therefore translation is just another way of defining a vector or a one-dimensional straight line. Locally speaking, it is easy to say what a vector is by giving some sense of direction and magnitude of the vector. But globally the universal expansion of space cannot be perceived as easily. Obviously, the universe contains near infinite mass. But for an observer outside of the universe, does he or she observe the universe as moving or stationary. For us, observers within the universe, we can assume that the linear momentum of the universe is zero (globally speaking). This assumption indirectly and implicitly defined an upper limit to all speeds and which from Special Relativity is the speed of light. Translational symmetry does not say anything about mass. It only says that, globally viewed, we can define the speed of light as a universal constant of nature. This is described by the principle of Lorentz invariance.

Rotational symmetry – Newton’s 2nd law of motion gives the definition of a force (also in the 1st law). This force gives a rate of change of velocity giving the definition of acceleration. All accelerations contain rotational symmetry. But angular momentum is the product of linear momentum and a metric while force is defined as the time rate of change of linear momentum. When force and metric are multiplied resulted a moment of force or torque. Locally, this gives us Planck’s constant. Globally, this symmetry becomes very complicated resulting in the use of tensor calculus which is just higher dimensional vector calculus. Globally, this symmetry is described by the Principle of Equivalence of Einstein’s General Relativity. Rotational symmetry does say something about mass. The Principle of Equivalence does give a hint for the existence of two type of mass. But the principle mistakenly identified these two masses as the inertial mass and the gravitational mass. The gravitational mass is the same mass viewed locally while the inertial mass is the same mass viewed globally or it might be the other way around (gravity mass is mass viewed globally and inertial mass is mass viewed locally). There is no different between the two. Both of them can be called potential mass. But there is another mass that which we now can be called as the kinetic mass. This is a mass of motion in contrast to inertial mass or potential mass that is a mass of rest.

Temporal symmetry - A friend of mine asked me a question: given enough time can we understand the truth of everything? Understanding is not the problem but time is. By our inherent biological clocks, we all know that time is real. From historical statistics, we came to accept our human average lifespan. Through the progress of civilization, humans endeavor to try to understand and control time. But now I know it takes no time at all to know and to accept our human limitations. We cannot know everything unless we know what time is. The truth is we will never know what time is. The irony is that all physical theories are based on the control of time: firstly, by assuming that it flows uniformly, evenly and constantly, secondly, by combining time to space resulting the concept of spacetime. Both are based on our assumption that time flows although we don’t really know that time flows. Time can be perceived as the rate of change of matter to energy or matter to space; energy to matter or energy to space; space to matter or space to energy. With each step of change, a probability is associated. For example: take the equation E=mc^2, says that a lot of energy can be derived from a small amount of mass. But the probability of this event is 1/c^2 otherwise everything would be energy and no matter at all. Conversely, m=E/c^2 says that a small amount of mass can be derived from a lot of energy and the not normalized probability is c^2. An equation relating continuous space to energy can be formulated: continuous space = cE. The constant c is again the speed of light. This equation says that a lot of space can be derived from energy but the probability is 1/c. Then space can be indirectly derived from mass by continuous space = mc^3. A small amount of mass can give an even greater amount of space but the not normalized probability is 1/c^3. This implied that it is even more probable for mass to change into energy than to continuous space. All these changes implied a broken symmetry for time.

Because of time broken symmetry, globally speaking the universe is composed of matter only while at the local infinitesimal domain only can one finds the existence of antimatter or created from high-energy experiments that takes the above mention probabilities into consideration. It is also the cause of time broken symmetry that makes the structure of atom the way it is: the mass of the proton very much larger than the mass of the electron resulting in a stable atom.

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# Time as a Broken Symmetry

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