Composing Energy-mass and Space-time gives Einstein’s Identity
- The origin in the above diagram is the Space-time Identity, that is \(c^2\). Einstein’s Identity connects Energy-mass as another symmetry with \(c^2\) as the Identity.
The Identity by Einstein \(E=mc^2\) is seen as a composite of Energy-Mass Duality and Space-Time Duality, where \(E\) is Energy, \(m\) is mass and \(c\) is the Speed of Light. We rearrange the equation as
\[\frac{E}{m} = c \cdot c\]Recognizing the multiplication operation (\(\cdot\)) as \(compose^i\) and the division operation (\(-\)) as \(compose_i\), that is, multiplication is identified as dual of division, we see
\[(Energy)\,\, compose_i \,\,(mass) = (Speed of Light)\,\, compose^i \,\,(Speed of Light)\]The two expressions on either side of the “equal to sign” are recognized as reflections of each other. Bring them on the same side of \(=\) by Identifying them with the orientations \((-) compose^d\) and \(compose_d (+)\) (which are individually identity but duals to each other), to the two duals on the right hand side. They are brought to the left hand side side as:
\[(Energy)\,\, compose_i \,\,(mass)\,\,\,\, \left( \,\, (-)\,\, compose^d \,\,(Speed of Light) )\,\, compose^i \,\,( \,\,(Speed of Light) \,\, compose_d(+) \right) = 0\]Recognizing Speed of light as expression of Space-time Duality at equilibrium, that is \(c_0= \left( \,\,(Space)\,\, compose_i \,\,(time)\,\, \right)_0\) and \(c^0= \left( \,\, (Space)\,\, compose_i \,\,(time) \,\, \right)^0\), (in this case \(c_0=c^0=c\)), we see
\[(Energy)\,\, compose_i \,\, (mass) \,\,\,\, \left( \,\, (-)\,\, compose^d \,\, \left(\,\, (Space)\,\, compose_i \,\,(time) \,\,\right)_0 \,\, \right) \,\, compose^i \,\, \left( \,\, \left( (Space)\,\, compose_i \,\,(time) \,\, \right)^0 \,\, compose_d(+) \,\, \right) = 0\]In symbols, Recognize Energy as \(E\), mass as \(m\), Space as \(X\), time as \(t\), and compose as \(\circ\), we see
\[(E\,\, \circ_i \,\, m )\,\,\,\, \left( \,\, (-)\,\, \circ^d\,\, \left( \,\, (X)\,\, \circ_i\,\, (t) \,\, \right)_0 \,\, \right)\,\, \circ^i\,\, \left( \,\, \left(\,\,(X)\,\, \circ_i\,\, (t) \,\,\right)^0 \,\, \circ_d (+) \,\, \right) =0\]- For keeping track, in usual language, the above equation reads as \(\frac{E}{m} \left(-\,\, (\frac{X}{t})_{eq} \,\, (\frac{X}{t})_{eq} + \,\,\right) = 0\), where \((\frac{X}{t})_{eq} = c\).
Back to where we were, since composition here is associative, we may drop the brackets (paranthesis) one by one, starting from inner to outer, that is from the signs \((-)\) and \((+)\), we see
\[(E\,\, \circ_i \,\, m) \,\,\,\, \left(\,\, -\,\, \circ^d\,\, \left(\,\, (X)\,\, \circ_i\,\, (t) \,\, \right)_0 \,\, \right)\,\, \circ^i\,\, \left( \,\, \left( \,\,(X)\,\, \circ_i\,\, (t) \right)^0 \,\, \circ_d + \,\, \right) =0\]Now removing the brackets from \((X)\) and \((t)\), we see
\[(E\,\, \circ_i \,\, m) \,\,\,\, \left(\,\, -\,\, \circ^d\,\, \left(\,\, X\,\, \circ_i\,\, t \,\, \right)_0 \,\, \right)\,\, \circ^i\,\, \left( \,\, \left( \,\,X\,\, \circ_i\,\, t \right)^0 \,\, \circ_d + \,\, \right) =0\]Further peeling gives
\[E\,\, \circ_i \,\, m \,\,\,\, \,\, -\,\, \circ^d\,\, \left(\,\, X\,\, \circ_i\,\, t \,\, \right)_0 \,\, \,\, \circ^i\,\, \,\, \left( \,\,X\,\, \circ_i\,\, t \right)^0 \,\, \circ_d + \,\, =0\]In Dirac’s Bra-Ket notation, this is
\[<\,\,E\,\, | \,\, m\,\,> \,\,\,\, <\,\, -\,\,| \,\, \langle \,\, \langle\,\, X\,\, |\,\, t \,\, \rangle_0 \,\, \,\, |\,\, \,\, \langle \,\,X\,\, | \,\, t \,\, \rangle^0 \,\, \rangle \,\,| \,\, + \,\, > =0\]This is the most reduced form which is equivalent to \(\left(\frac{E}{m}\right)(-c \cdot c+)=0\) or \(\frac{E}{m}-c \cdot c+=0\).
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Starting with the reduced form above which is equivalent to \(c^2-c^2=0\), we may reverse the steps to obtain a “composition” which is the Einstein’s Identity.
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This is interpreted as: Energy-mass Duality and Space-time Duality are dual to each other. The Energy-mass duality axis is Orthogonal to the Space-time Duality and they meet only at the Light Cone which is where Space and Time are at equilibrium with each other. Hence \(E=mc^2\) is a two Phase Equilibrium between (Energy-mass) as one Phase and (Space-time) as another Phase. It may be reduced further to Energy, mass, Space (\(X = (x_1,x_2,x_3)\,\,\)) and time as its components.
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The Space-time Metric is what decides the amount (measure) of how far from Equilibrium the Space-time fabric is. The Light Cone in the resulting Phase Diagram gives the Equilibrium Phase Boundary between Space and Time as the Phases. Einstein’s Field Equations are the corresponding Maxwell’s relations for the Thermodynamics of Energy-mass and Space-time.
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We see that \(E = mc^2\) is a special case (Equilibrium) of a general Thermodynamic Potential Energy \(E\), which may be defined as \(E = mc^2 \,\,-x_1^2-x_2^2-x_3^2 + c^2t^2\)
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In the Language of Group Theory: \(c^2\) is the identity, \(-\) (Division) is the group operation. Energy (\(E\)) and mass (\(m\)) are inverses of each other with ; Space (\(X\)) and time (\(t\)) are inverses of each other.
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That is, \(\frac{E}{m}= c^2\) and \(\frac{X}{t}= c^2\). If we Identify \(c^2\), the identity with the symbol \(e\), then this reads \(\frac{E}{m}= e\) and \(\frac{X}{t}= e\). That is, Energy and mass are reflections of each other and Space and time are reflections of each other.
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The other operation \(+\) with 0 as Identity allows defining Energy to be the other identity. This makes the group into a Field.
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The above representation as a group only happens at equilibrium. When there is no equilibrium, the measure of how far from equilibrium is given by the Thermodynamic expression \(E= mc^2-x_1^2-x_2^2-x_3^2 + c^2t^2\), where \(E\) acts as a functional on \(m\) and the metric (-,-,-,+) acts as the Integral to compose space and time as duals. This composition of a Group with addition and subtraction to give a thermodynamic type scalar quantity, and multiplication to give inverses with \(c^2\) as the Identity can be represented as a Field. A field allows addition and subtraction; multiplication and division. Thus, Energy-mass and Space-time can be seen as corresponding to Field of Complex Numbers (Provided all three dimensions of space are identified as one). In other words, Complex Numbers represent the Phase Diagram of Energy-mass and Space-time with the axis as their Phase boundaries.
- The above may seem to be dimensionally wrong but I request us to be patient and Identify all forms of energy as mass and all forms of distance as mass as well to see all equations as an Equilibrium Mass Balance.
If Energy and Mass are identified, and Space and Time are Identified, the above graph of a two dimensional Euclidean Plane is seen as two Circles orthogonal to each other about a common axis. This represents the Globe, with Space-Time as the Equator and Energy-Mass as the Prime Meridian. Their meeting point at the South Pole as the Identity which is called \(c^2\), whereas the North Pole to be the dual Identity.
Identifying Identity as Equilibrium
- Note that writing \(=0\) at the end of lines in the scalar equations of energy is for completeness of mentioning it as the identity or to represent that the left hand expression is in equilibrium. This shows that assuming existence of identity (0) is not needed, we can simply call the expression to be balanced, but we choose to do so as a pointer that all equilibriums are forms of expression of a common abstract identity.