Form Invariance: Searching for Laws of Physics
How can we be guided in the search for laws of physics by the form that the mathematical statements must take? What is the special significance in this search of 4-vector and 4-tensor equations?
QUESTIONS
(Score is number right minus number wrong.)
In the super/subscript notation on the far left-hand-side of Equations (1) and (2) used to represent elements of the matrices on the right-hand-sides, the
superscript is the row index, the subscript the column index.
superscript is the column index, the subscript is the row index.
The correct pattern for the transform of a electromagnetic current 4-vector from the unprimed Lorentz frame to the primed Lorentz frame is given by,
Equation (5a).
Equation (5b).
Equation (5c).
Equation (5d).
The correct pattern for the transform of a electromagnetic field 4-tensor from the unprimed Lorentz frame to the primed Lorentz frame is given by,
Equation (6a).
Equation (6b).
Equation (6c).
Equation (6d).
In Equation (7a),
The connecting index is mu, the dummy index is nu
The connecting index is nu, the dummy index is mu.
Both mu and nu are connecting indices.
Both mu and nu are dummy indices.
Equation (7a) is, in reality, a compact statement for
one equation.
two separate equations.
three separate equations.
four separate equations.
Equation (8) is, in reality, a compact statement for
one equation.
two separate equations.
three separate equations.
four separate equations.
Equation (8) transforms Equation (7a) into Equation (7b), i.e. Equation (9) is
True.
False.
What Equation (9) represents is simply multiplying each of four equations by an element of the matrix in Equation (1).
True.
False.
What Equation (9) represents is simply multiplying each of four equations by a different element of the matrix in Equation (1) and then summing the four resulting equations to form a new equation. This is repeated in four different ways to create four new equations.
True.
False.
To further the transformation begun in Equation (9) into quantities measured in the primed coordinates system, we should substitute into Equation (9),
Equation (10a).
Equation (10b).
Equation (10c).
Equation (10d).
To yet further the transformation begun in Equation (9) into quantities measured in the primed coordinates system, we should subsitute into Equation (9),
Equation (11a).
Equation (11b).
Equation (11c).
Equation (11d).
Equation (12) correctly transforms all quantities in Equation (7a) measured in the unprimed system into quantities measured in the primed system.
True
False
Further simplification can be made using Equation (3). In this case, we introduce two Kronecker deltas, one of which is
Equation (13a).
Equation (13b).
Equation (13c).
Both Equation (7a) and Equation (7b) are correct, each in terms of quantities measured in different Lorentz frames.
True
False
The dummy index in Equation (14) is
sigma.
epsilon.
delta.
The sum on the left-hand-side of Equation (14) is correctly reduced by
Equation (14a).
Equation (14b).
Equation (14c).
CONCLUSION: Laws of physics expressed in 4-vector (and 4-tensor) equations will be guaranteed to have the same mathematical form in different Lorentz frames. Compare Equation (7a) and Equation (7b). Thus, if we are trying to find laws of physics that are consistent with Special Relativity, we should formulate them in 4-vector or 4-tensor equations.
EQUATIONS
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