In coordinates (t , x), a contravariant second rank tensor A has non-zero diagonal
components Att = P and Axx = 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) → (t′, x′) to another frame that moves with relative speed v in
the +x direction, so that A → A′. If A′tt and A′xx are the diagonal components of
A′, then setting the speed of light c = 1, and with 𝛾 =
1
√1-v2 , which of the
following option(s) is/are correct?
Options:
(A) A′tt = 𝛾2P + 𝛾2v2Q
(B) A′tt = 𝛾2v2P + v2Q
(C) A′xx = 𝛾2v2P + 𝛾2Q
(D) A′xx = v2P + 𝛾2Q
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.50)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Mathematical Physics** within *Section 1: Mathematical Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Mathematical Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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