A Quick Review of Special Relativity in Visual Studio .NET

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A Quick Review of Special Relativity
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Fundamentally, our commonsense intuition about how the universe works is tied up in notions about space and time. In 1905, Einstein stunned the physics world with the special theory of relativity, a theory of space and time that challenges many of these closely held commonsense assumptions about how the world works. By accepting that the speed of light in vacuum is the same constant value for all observers, regardless of their state of motion, we are forced to throw away basic ideas about the passage of time and the lengths of rigid objects. This book is about the general theory of relativity, Einstein s theory of gravity. Therefore our discussion of special relativity will be a quick overview of concepts needed to understand the general theory. For a detailed discussion of special relativity, please see our list of references and suggested reading at the back of the book. The theory of special relativity has its origins in a set of paradoxes that were discovered in the study of electromagnetic phenomena during the nineteenth
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Special Relativity
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century. In 1865, a physicist named James Clerk Maxwell published his famous set of results we now call Maxwell s equations. Through theoretical studies alone, Maxwell discovered that there are electromagnetic waves and that they travel at one speed the speed of light c. Let s take a quick detour to get a glimpse into the way this idea came about. We will work in SI units. In careful experimental studies, during the rst half of the nineteenth century, Ampere deduced that a steady current J and the magnetic eld B were related by B = 0 J (1.1)
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However, this law cannot be strictly correct based on simple mathematical arguments alone. It is a fundamental result of vector calculus that the divergence of any curl vanishes; that is, A =0 for any vector A. So it must be true that B =0 (1.3) (1.2)
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However, when we apply the divergence operator to the right-hand side, we run into a problem. The problem is that the continuity equation, which is the mathematical description of the conservation of charge, tells us that + J =0 t (1.4)
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where is the current density. Therefore, when we apply the divergence operator to the right-hand side of (1.4), we obtain 0 J = 0 J = 0 t (1.5)
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We can take this process even further. Gauss s law tells us how to relate the charge density to the electric eld. In SI units, this law states E= 1 0 (1.6)
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Special Relativity
This allows us to rewrite (1.5) as 0 E = 0 0 E = 0 0 t t t (1.7)
Putting our results together, we ve found that B = 0 0 E t (1.8)
when in fact it must be zero. Considerations like these led Maxwell to x up Ampere s law. In modern form, we write it as B = 0 J + 0 0 E t (1.9)
E The extra term 0 0 t is called the displacement current and its presence led to one of Maxwell s most dramatic discoveries. Using simple vector calculus, one can show that the electric and magnetic elds satisfy the following wave equations:
2 E = 0 0 Now, the wave equation is
2 E t 2
2 B = 0 0
2 B t 2
2 f =
1 2 f v 2 t 2
where v is the velocity of the wave. Comparison of these equations shows that electromagnetic waves in vacuum travel at speed v= 1 = 3 108 m/s = c 0 0
where c is nothing more than the speed of light. The key insight to gain from this derivation is that electromagnetic waves (light) always travel at one and the same speed in vacuum. It doesn t matter who you are or what your state of motion is, this is the speed you are going to nd. It took many years for this insight to sink in and it was Einstein who simply accepted this result at face value.
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