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Chapter 12: 50. (page 568)

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if is symmetric, show that is also symmetric, and likewise for antisymmetric).

Short Answer

Expert verified

The symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation.

Step by step solution

01

Expression for the property of symmetric and antisymmetric tensor:

Write the property of the symmetric tensor.

t渭谓=t谓渭

Write the property of an anti-symmetric tensor.

role="math" localid="1650574388909" t渭谓=-t谓渭

Here, a negative sign for anti-symmetric tensor.

02

Determine the symmetry or antisymmetry of a tensor:

Consider

t-魏位=A渭魏A谓位t渭谓

Here, A is the Lorentz transformation matrix.

Since 渭and v both are summed from 0 to 3, both the values can be interchanged.

Hence, the above equation becomes,

t-魏位=A渭魏A谓位t渭谓t-魏位=A谓位A渭魏t谓渭t-魏位=A渭魏A渭位(-t渭谓)t-魏位=卤t-魏位

Therefore, the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation.

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Most popular questions from this chapter

鈥淚n a certain inertial frame S, the electric field E and the magnetic field B are neither parallel nor perpendicular, at a particular space-time point. Show that in a different inertial system S, moving relative to S with velocity v given by

v1+v2/c2=E脳BB2+E2/c2

the fieldsEandBare parallel at that point. Is there a frame in which the two are perpendicular?

Generalize the laws of relativistic electrodynamics (Eqs. 12.127 and 12.128) to include magnetic charge. [Refer to Sect. 7.3.4.]

Use the Larmor formula (Eq. 11.70) and special relativity to derive the Lienard formula (Eq. 11. 73).

P=渭0q2a26蟺c鈥夆赌夆赌�(11.70)P=渭0q2纬66蟺c(a2-|蠀脳ac|2)鈥夆赌夆赌�(11.73)

Define proper acceleration in the obvious way:

伪渭=d畏渭d蟿=d2x渭d蟿2

(a) Find伪0and 伪 in terms of u and a (the ordinary acceleration).

(b) Express伪渭伪渭in terms of u and a.

(c) Show that畏渭伪渭=0.

(d) Write the Minkowski version of Newton鈥檚 second law, in terms of伪渭. Evaluate the invariant productK渭畏渭.

12.48: An electromagnetic plane wave of (angular) frequency 蝇is travelling in the xdirection through the vacuum. It is polarized in the ydirection, and the amplitude of the electric field is Eo.

(a) Write down the electric and magnetic fields, role="math" localid="1658134257504" E(x,y,z,t)and B(x,y,z,t)[Be sure to define any auxiliary quantities you introduce, in terms of 蝇, Eo, and the constants of nature.]

(b) This same wave is observed from an inertial system S鈫�moving in thexdirection with speed vrelative to the original system S. Find the electric and magnetic fields in S鈫�, and express them in terms of the role="math" localid="1658134499928" S鈫�coordinates: E(x鈫�,y鈫�,z鈫�,t鈫�)and B(x鈫�,y鈫�,z鈫�,t鈫�). [Again, be sure to define any auxiliary quantities you introduce.]

(c) What is the frequency 蝇鈫�of the wave in S鈫�? Interpret this result. What is the wavelength 位鈫�of the wave in S鈫�? From 蝇鈫�and 位鈫�, determine the speed of the waves in S鈫�. Is it what you expected?

(d) What is the ratio of the intensity in to the intensity in? As a youth, Einstein wondered what an electromagnetic wave would like if you could run along beside it at the speed of light. What can you tell him about the amplitude, frequency, and intensity of the wave, as approaches ?

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