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SPECIAL RELATIVITY Background (Problems with Classical Physics) Classical mechanics are valid at low speeds But are invalid at speeds close to the speed of light

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Special Relativity (Background) a special case of the general theory of relativity for measurements in reference frames moving at constant velocity. predicts how measurements in one inertial frame appear in another inertial frame. How they move wrt to each other.

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Reference Frames The problems described will be done using reference frames which are just a set of space time coordinates describing a measurement. eg. z y x t

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Reference Frames We therefore first review Newtonian mechanics using inertial frames. NB: This is not a foreign concept since any physical event must be wrt to some frame of reference. eg. a lab.

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Galilean-Newtonian Relativity According to the principle of Newtonian Relativity, the laws of mechanics are the same in all inertial frames of reference. i.e. someone in a lab and observed by someone running.

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Galilean-Newtonian Relativity Galilean Transformations

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allow us to determine how an event in one inertial frame will look in another inertial frame. assume that time is absolute.

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Galilean Transformations In S an event is described by (x,y,z;t). How does it look in S′? z x t z′ x′ t′ utx′ u O′O

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Galilean Transformations For Galilean transforms t = t′ From the diagram, And z x t z′ x′ t′ utx′ u O′O

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Galilean Transformations Velocities can also be transformed. Using the previous equations we, (addition law for velocities)

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Galilean Transformations Acceleration can also be transformed! When we do we get, Thus Force (F=ma) is same in all inertial frames.

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Galilean Transformations Transforming Lengths

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Galilean Transformations How do lengths transforms transform under a Galilean transform?

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Galilean Transformations How do lengths transforms transform under a Galilean transform? Note: to measure a length two points must be marked simultaneously.

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Galilean Transformations Consider the truck moving to the right with a velocity u. Two observers, one in S and the other S′ measure the length of the truck. U S′S XAXA XBXB

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Galilean Transformations In the S frame, an observer measures the length = X B -X A In the S′ frame, an observer measures the length = X′ B -X′ A U S′S XAXA XBXB

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Galilean Transformations each point is transformed as follows: U S′S XAXA XBXB

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Galilean Transformations U S′S XAXA XBXB 0 Therefore we find that 0 Since

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Galilean Transformations Hence for a Galilean transform, lengths are invariant for inertial reference frames.

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Summary (Important consequence of a Galilean Transform) All the laws of mechanics are invariant under a Galilean transform.

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Problems with Newtonian- Galilean Transformation Are all the laws of Physics invariant in all inertial reference frames?

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Problems with Newtonian- Galilean Transformation Are all the laws of Physics invariant in all inertial reference frames? For example, are the laws of electricity and magnetism the same?

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Problems with Newtonian- Galilean Transformation Are all the laws of Physics invariant in all inertial reference frames? For example, are the laws of electricity and magnetism the same? For this to be true Maxwell's equations must be invariant.

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Problems with Newtonian- Galilean Transformation From electromagnetism we know that, Since and are constants then the speed of light is constant.

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Problems with Newtonian- Galilean Transformation From electromagnetism we know that, Since and are constants then the speed of light is constant. However from the addition law for velocities

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Problems with Newtonian- Galilean Transformation Therefore we have a contradiction! Either the additive law for velocities and hence absolute time is wrong Or the laws of electricity and magnetism are not invariant in all frames.

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