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  3. 1 0 0 0 let 0010 let l4 x 4...

Question: 1 0 0 0 let 0010 let l4 x 4...

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-1 0 0 0 Letη 0010 Let L-(4 x 4 matrices A such that ATpA a) Show that if A, BE L then AB E L b) Show that if A E C then A-1 E L c) Suppose Sally declares herself to be at the center of the universe, and sets her watch to 0. In physics, this is an example of an event. By definition then, this event occurred at time 0, and position 0 (0,0,0), according to Sally. This information can be encoded as a vector O- in IR Suppose sometime later, a bolt of lightning strikes a house. This is another event, which Sally records as happening at time t, and Cartesian coordinates (z, y, z) This information can be encoded as a vector V -- inR 2 The spacetime-displacement between these events is v - OV - The space-time interval between events O and V is defined to be s = v,v Evaluate s d) Not everyone agrees with Sallys contentions. Observers in different places can declare themselves to be at the center of the universe, set their watches to 0 at different times, and have their co-ordinate axes rotated relative to Sallys. In special relativity, observers moving relative to one another dont even agree on how fast clocks tick, or the distance between two events. Suppose another observer, John, records the spacetime-displacement between Sally setting her watch to 0, and the lightning striking the house as The laws of physics tell us that v and v are always related by a matrix transfor- mation, v,-Λν, where Λ L Let the space-time separation according to John be given by s-(vTv Show that s- s

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