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Chapter 12: Q12.50P (page 564)

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if t渭vis symmetric, show thatt炉渭v 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渭惫=t惫渭

Write the property of an anti-symmetric tensor.

t渭惫=-t惫渭

Here, a negative sign for anti-symmetric tensor.

02

Determine the symmetry or antisymmetry of a tensor:

Considert鈭�K位=蚂渭K蚂v位t渭v

Here,蚂 is the Lorentz transformation matrix.

Since渭 and v both are summed from0 to 3, both the values can be interchanged.

Hence, the above equation becomes,

t鈭�位K=蚂渭K蚂渭Kt渭vt鈭�位K=蚂v位蚂渭Ktv渭t鈭�位K=蚂渭K蚂渭位(鈭�t渭v)t鈭�位K=卤t鈭�K位

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

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Most popular questions from this chapter

(a) What鈥檚 the percent error introduced when you use Galileo鈥檚 rule, instead of Einstein鈥檚, withvAB=5mi/handvBC=60mi/hand?

(b) Suppose you could run at half the speed of light down the corridor of a train going three-quarters the speed of light. What would your speed be relative to the ground?

(c) Prove, using Eq. 12.3, that ifvAB<candvBC<cthenvAC<cInterpret this result.

Find the velocity of the muon in Ex. 12.8.

(a) Draw a space-time diagram representing a game of catch (or a conversation) between two people at rest, apart. How is it possible for them to communicate, given that their separation is spacelike?

(b) There's an old limerick that runs as follows:

There once was a girl named Ms. Bright,

Who could travel much faster than light.

She departed one day,

The Einsteinian way,

And returned on the previous night.

What do you think? Even if she could travel faster than light, could she return before she set out? Could she arrive at some intermediate destination before she set out? Draw a space-time diagram representing this trip.

Show that

k渭k渭=胃鈥�(u2c2)cos21-u2c2

Where胃is the angle between u and F.

Check Eq. 12.29, using Eq. 12.27. [This only proves the invariance of the scalar product for transformations along the x direction. But the scalar product is also invariant under rotations, since the first term is not affected at all, and the last three constitute the three-dimensional dot product a-b . By a suitable rotation, the x direction can be aimed any way you please, so the four-dimensional scalar product is actually invariant under arbitrary Lorentz transformations.]
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