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GATE 2025 PH – Question 50

Mathematical Physics · tensors: tensor transformations, covariant and contravariant tensors. · 2 marks · Multiple select

In coordinates $(t,x)$, a contravariant second rank tensor has $A^{tt}=P,A^{xx}=Q$, all other entries zero. A Lorentz boost in +x with speed $v$ gives $A\to A^{\prime}$. Set $c=1$ and $\gamma=(1-v^2)^{-1/2}$. Which option(s) is/are correct?

  1. $A^{\prime tt}=\gamma^2P+\gamma^2v^2Q$
  2. $A^{\prime tt}=\gamma^2v^2P+v^2Q$
  3. $A^{\prime xx}=\gamma^2v^2P+\gamma^2Q$
  4. $A^{\prime xx}=v^2P+\gamma^2Q$

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Correct answer: (A) $A^{\prime tt}=\gamma^2P+\gamma^2v^2Q$; (C) $A^{\prime xx}=\gamma^2v^2P+\gamma^2Q$

Explanation

A contravariant rank-2 tensor transforms as
$$A^{\prime\mu\nu}=\Lambda^\mu{}_\alpha\Lambda^\nu{}_\beta A^{\alpha\beta},$$
with the boost matrix (in $(t,x)$, $c=1$)
$$\Lambda=\begin{pmatrix}\gamma&-\gamma v\\-\gamma v&\gamma\end{pmatrix}\quad\text{(for the sign convention of a boost along }+x\text{)}.$$

Only $A^{tt}=P$ and $A^{xx}=Q$ are non-zero, so
$$A^{\prime tt}=\Lambda^t{}_t\Lambda^t{}_tA^{tt}+\Lambda^t{}_x\Lambda^t{}_xA^{xx}=\gamma^2P+\gamma^2v^2Q,$$
$$A^{\prime xx}=\Lambda^x{}_t\Lambda^x{}_tA^{tt}+\Lambda^x{}_x\Lambda^x{}_xA^{xx}=\gamma^2v^2P+\gamma^2Q .$$

These match options **A** and **C**. (The off-diagonal component $A^{\prime tx}=-\gamma^2v(P+Q)$ is also generated, but it is not asked about.)