- #1
lalbatros
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In the story below, where would you see possible alternatives, or where would you see a problem?
(01) Let us consider a set of physicists {P0, P1, P2, P3, ...} each at rest in their own inertial frames.
(02) Let us elect one of them (P0) as the boss to manage an experiment.
(03) Let us assume the velocities of each physicist with respect to the boss are {v1, v2, v3, ...}.
(04) The boss asks each physicist to measure in his own frame the spacetime coordinates E(x,t) of a large number of events he has scheduled, something like sparkles in a firework.
(05) All the physicists are using the same measuring instruments.
(06) All physicist "zero" their coordinate systems on the origin of the boss coordinate system.
(07) All the physicists must return their result to the boss for analysis.
(08) The boss is now looking for a relation between all these measurements.
(09) The boss choses for a linear relation between the coordinates measured by all physicists.
(10) The boss assumes this linear relation depends on the velocities of his colaborators.
(11) In summary he assumes: E1=T(v1)E0, E2=T(v2)E0, where T(v) is a 2x2 matrix depending on v
(12) The boss organizes a meeting with his collaborators.
(13) He asks them: "What should be the properties of these matrices T(vi)"?
(14) One of them, called René, writes these properties on the black board:
(15) >> a. T(0) should be the unit matrix such that T(0)Ei=Ei
(16) >> b. Inverse(T(v))=T(-v) to assure forward and backward transformations are similar
(17) >> c. T(vi).T(vj) should also be a transformation matrix, or in other words:
(18) >> d. T(vi).T(vj)=T(vij) without prejudice about how vij should be calculated
(19) >> He concludes saying:
(20) The set of matrices T(v) is a one-parameter matrix group.
(20) The parameter involved is the relative velocity v.
Assume you are one of the other physicists and you are invited to express your opinion:
Q1- What do you think about the overall experiment?
Q2- Is the experimental protocol correctly defined?
Q3- What do you think about the properties suggested by René?
Q4- Do you think all properties should be needed?
Q5- Do you think each property has a physical meaning? And which one?
Q6- Do you agree with each physical meaning?
Q7- Do you think these properties might not all be verified experimentally?
Q8- In summary, according to you, should T(v) be a one-parameter matrix group?
Assuming the conclusion by René is right:
Q9- Do you think there are many different such groups (one-parameter 2x2 matrix group)
Q10- How many could you lists?
Q11- Are some of them already familiar to you?
Q12- To finish the job of the boss, which group would you suggest?
Q13- Write a formula for T(v).
Q14- Would you have ideas for further experiments?
(01) Let us consider a set of physicists {P0, P1, P2, P3, ...} each at rest in their own inertial frames.
(02) Let us elect one of them (P0) as the boss to manage an experiment.
(03) Let us assume the velocities of each physicist with respect to the boss are {v1, v2, v3, ...}.
(04) The boss asks each physicist to measure in his own frame the spacetime coordinates E(x,t) of a large number of events he has scheduled, something like sparkles in a firework.
(05) All the physicists are using the same measuring instruments.
(06) All physicist "zero" their coordinate systems on the origin of the boss coordinate system.
(07) All the physicists must return their result to the boss for analysis.
(08) The boss is now looking for a relation between all these measurements.
(09) The boss choses for a linear relation between the coordinates measured by all physicists.
(10) The boss assumes this linear relation depends on the velocities of his colaborators.
(11) In summary he assumes: E1=T(v1)E0, E2=T(v2)E0, where T(v) is a 2x2 matrix depending on v
(12) The boss organizes a meeting with his collaborators.
(13) He asks them: "What should be the properties of these matrices T(vi)"?
(14) One of them, called René, writes these properties on the black board:
(15) >> a. T(0) should be the unit matrix such that T(0)Ei=Ei
(16) >> b. Inverse(T(v))=T(-v) to assure forward and backward transformations are similar
(17) >> c. T(vi).T(vj) should also be a transformation matrix, or in other words:
(18) >> d. T(vi).T(vj)=T(vij) without prejudice about how vij should be calculated
(19) >> He concludes saying:
(20) The set of matrices T(v) is a one-parameter matrix group.
(20) The parameter involved is the relative velocity v.
Assume you are one of the other physicists and you are invited to express your opinion:
Q1- What do you think about the overall experiment?
Q2- Is the experimental protocol correctly defined?
Q3- What do you think about the properties suggested by René?
Q4- Do you think all properties should be needed?
Q5- Do you think each property has a physical meaning? And which one?
Q6- Do you agree with each physical meaning?
Q7- Do you think these properties might not all be verified experimentally?
Q8- In summary, according to you, should T(v) be a one-parameter matrix group?
Assuming the conclusion by René is right:
Q9- Do you think there are many different such groups (one-parameter 2x2 matrix group)
Q10- How many could you lists?
Q11- Are some of them already familiar to you?
Q12- To finish the job of the boss, which group would you suggest?
Q13- Write a formula for T(v).
Q14- Would you have ideas for further experiments?
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