Understanding the Covariant Derivative of T_αβT^αβ in Relativistic Physics

The Hilbert stress-energy tensor, denoted by T_αβ, is a fundamental concept in general relativity. It represents the distribution of matter and energy in spacetime. A common question arises regarding the behavior of the covariant derivative of this tensor: specifically, whether nabla_μ T_{αβ} T^{αβ} is zero.

Key Concepts and Definitions

In general relativity, the stress-energy tensor T_αβ is a (0,2) tensor that describes the distribution of energy, momentum, and stress at each point in spacetime. One of the most important properties of the stress-energy tensor is its divergence being zero, i.e., nabla_μ T^{μν} 0 . This is a consequence of the conservation of energy and momentum in curved spacetime.

The Conjecture and Analysis

The statement that nabla_μ T_{αβ} T^{αβ} 0 is generally incorrect. Although the stress-energy tensor itself is divergence-free, its covariant derivative does not always vanish. This can be understood by analyzing the properties of the trace of the stress-energy tensor, which is given by T^{αβ} T_{αβ} .

Derivation and Counterexample

Let's start from the expression nabla_μ T_{αβ} T^{αβ} and expand it step by step:

Using the metric tensor g_{αγ} and g_{βδ} , we can rewrite the expression:

begin{align} nabla_μ T_{αβ} T^{αβ} g_{αγ} g_{βδ} nabla_μ T^{γδ} T^{αβ} - T_{αβ} nabla_μ T^{αβ} end{align}

Since the metric tensor is covariantly constant ( nabla_μ g_{αβ} 0 ), the first term simplifies to:

begin{align} nabla_μ T_{αβ} T^{αβ} T_{αβ} nabla_μ T^{αβ} end{align}

Given the fact that the stress-energy tensor can be written as T_{αβ} T_{(αβ)} - frac{1}{2} g_{αβ} T^{μν} T_{μν} , where T_{(αβ)} is the traceless part and T^μν T_μν is the scalar part, we can further simplify:

begin{align} nabla_μ T_{αβ} T^{αβ} T_{αβ} nabla_μ T^{αβ} end{align}

For a pressureless perfect fluid, the stress-energy tensor can be written as T_{αβ} ρ^2 g_{αβ} , where ρ is the energy density. Therefore, the scalar part is:

begin{align} nabla_μ T_{αβ} T^{αβ} 2 ρ^2 nabla_μ ρ end{align}

This expression clearly shows that nabla_μ T_{αβ} T^{αβ} is not zero, since ρ is a scalar and the covariant derivative of a scalar field is generally non-zero unless the scalar field is itself constant.

Simplified Analysis and Concluding Remarks

The key takeaway is that, while the stress-energy tensor itself satisfies the divergence-free condition, the covariant derivative of its square term is not zero in general. This is due to the properties of the tensor itself and the nature of the metric in general relativity.

In summary, the correct expression is:

begin{align} nabla_μ T_{αβ} T^{αβ} 2 T_{αβ} nabla_μ T^{αβ} end{align}

This result highlights the importance of understanding the distinction between the divergence-free condition and the behavior of the covariant derivative of tensorial expressions.

Key Points Recap

The stress-energy tensor T_αβ is divergence-free: nabla_μ T^{μν} 0 . The expression nabla_μ T_{αβ} T^{αβ} is not zero in general due to the non-vanishing covariant derivative of the tensor. For a pressureless perfect fluid, the expression simplifies to 2 ρ^2 nabla_μ ρ , which is generally non-zero.

Related Keywords

Covariant Derivative: A generalization of the ordinary derivative to tensor fields on a manifold.

Stress-Energy Tensor: A (0,2) tensor that describes the distribution of energy, momentum, and stress in spacetime.

Hilbert Tensor: The stress-energy tensor used in general relativity.

Divergence-Free: A tensor field whose divergence is zero, indicating conservation laws in the context of general relativity.

Relativity: The theory that describes the interplay between space, time, and the force of gravity.