Question

In coordinates \( (t, x) \), a contravariant second rank tensor \( A \) has non-zero diagonal components \( A^{tt} = P \) and \( A^{xx} = Q \), with all other components vanishing, and \( P, Q \) being real constants. Here, \( t \) is time and \( x \) is space coordinate. Consider a Lorentz transformation \( (t, x) \to (t', x') \) to another frame that moves with relative speed \( v \) in the \( +x \) direction, so that \( A \to A' \). If \( A'^{tt} \) and \( A'^{xx} \) are the diagonal components of \( A' \), then setting the speed of light \( c = 1 \), and with \( \gamma = \frac{1}{\sqrt{1 - v^2}} \), which of the following option(s) is/are correct?

• A. \( A'^{tt} = \gamma^2 P + \gamma^2 v^2 Q \)
• B. \( A'^{tt} = \gamma^2 v^2 P + v^2 Q \)
• C. \( A'^{xx} = \gamma^2 v^2 P + \gamma^2 Q \)
• D. \( A'^{xx} = v^2 P + \gamma^2 Q \)

Hint:

When transforming the components of a second-rank tensor, the Lorentz transformation matrix must be applied to both time and space components, accounting for off-diagonal terms.

Answer 1

The Correct Option is A, C

Step 1: The transformation of a second-rank contravariant tensor under a Lorentz transformation is given by: \[ A'_{\mu\nu} = \Lambda_{\mu}^{\ \alpha} \Lambda_{\nu}^{\ \beta} A_{\alpha\beta}, \] where \( \Lambda_{\mu}^{\ \alpha} \) is the Lorentz transformation matrix. 
Step 2: For the diagonal components \( A^{tt} \) and \( A^{xx} \), we use the Lorentz transformation for each component: \[ A'^{tt} = \gamma^2 A^{tt} + \gamma^2 v^2 A^{xx}, \] \[ A'^{xx} = \gamma^2 v^2 A^{tt} + \gamma^2 A^{xx}. \] Substituting \( A^{tt} = P \) and \( A^{xx} = Q \), we get: \[ A'^{tt} = \gamma^2 P + \gamma^2 v^2 Q, \] \[ A'^{xx} = \gamma^2 v^2 P + \gamma^2 Q. \]