Problem 13 - A First Course in General Relativity

Question

The inequalities \(\left|T^{00}\right| \gg\left|T^{0 i}\right| \gg\left|T^{i j}\right|\) for a Newtonian system are illustrated in Exers. \(2(\mathrm{c})\). Devise physical arguments to justify them in general.

Step-by-Step Solution

Answer

The inequalities reflect that energy density (\(T^{00}\)) dominates due to mass, momentum densities (\(T^{0i}\)) are smaller due to low speeds, and stresses (\(T^{ij}\)) are smallest in Newtonian systems.

Step 1: Understanding the Inequalities

The inequalities \(\left|T^{00}\right| \gg\left|T^{0 i}\right| \gg\left|T^{i j}\right|\) describe the relative magnitudes of different components of the energy-momentum tensor \(T\) in a Newtonian system. Here, \(T^{00}\) represents energy density, \(T^{0i}\) are momentum densities, and \(T^{ij}\) are pressure or stress components. Understanding what each component represents is key to justifying their relative magnitudes based on physical conditions.

Step 1: Analyzing \(T^{00}\)

In a Newtonian system, which typically deals with low-speed and low-pressure conditions relative to the speed of light, the energy density, \(T^{00}\), is predominantly due to rest mass energy. Since mass is a primary carrier of energy in such systems, the term describing energy density tends to dominate over other forms of energy or momentum densities, making \(\left|T^{00}\right|\) the largest component.

Step 2: Evaluating \(T^{0i}\)

The components \(T^{0i}\) describe momentum densities. In most Newtonian systems, the speeds involved are much smaller than the speed of light, meaning the momentum given by mass times velocity \((p=mv)\) is quite small relative to the rest mass energy. Thus, while significant, \(\left|T^{0i}\right|\) is much smaller than \(\left|T^{00}\right|\).

Step 3: Examining \(T^{ij}\)

The components \(T^{ij}\) represent pressure or stress. For a Newtonian system, pressures or stresses are inherently smaller because the internal motion or interaction energies are much less significant than the bulk rest mass energy. Therefore, \(\left|T^{ij}\right|\) is much smaller than both the momentum densities and the energy density, justifying the last inequality.

Key Concepts

Newtonian system

A Newtonian system refers to a set of physical conditions where the principles of classical mechanics, formulated by Isaac Newton, apply. These systems assume that objects move at low velocities compared to the speed of light and that the forces and interactions can be adequately described using Newton's laws of motion. This framework is primarily concerned with:

• Describing the motion of objects due to forces.
• Emphasizing low-speed scenarios where relativistic effects (considered in Einstein's theory) can be neglected.
• Focusing on the mass of objects as a dominant energy carrier.

By considering these factors, Newtonian systems provide a straightforward understanding of mechanics suitable for everyday phenomena, like driving a car or dropping a ball. These are scenarios where the mathematical complexities of relativity can be safely ignored, making calculations simpler and more intuitive.

energy density

Energy density, represented by the component \(T^{00}\) in the energy-momentum tensor, refers to the amount of energy stored in a given volume of space. In a Newtonian system, this component is primarily determined by mass energy due to the rest mass of particles. Therefore, the equation \(E = mc^2\) simplifies to focusing primarily on the mass when velocities are low:

• Dominated by rest mass energy, as objects do not travel at speeds approaching light.
• Other forms of energy like kinetic energy become negligible relative to rest mass energy.
• This makes \(\left|T^{00}\right|\), the energy density, substantially larger than the momentum and stress components.

Understanding energy density is crucial, as it's a fundamental factor dictating how much "stuff" an object contains and how it's distributed in space.

momentum density

Momentum density is represented by the components \(T^{0i}\) of the energy-momentum tensor. It describes how much momentum is carried by objects within a particular volume. Given a Newtonian system, the momentum density is expressed by the formula \(p = mv\), where:

• \(m\) = mass of the object (constant in Newtonian dynamics).
• \(v\) = velocity of the object.

The significance of momentum density in a Newtonian realm is moderated by the object's speed, which is considerably less than the speed of light. Because of this, even if objects have mass and thus energy, their momentum relative to mega-energetic phenomena (like light speed) remains small:

• Momentum density \(\left|T^{0i}\right|\) is considerably smaller than energy density \(\left|T^{00}\right|\).
• Velocity influences the momentum, so slower movements ensure momentum is modest by comparison.

This makes analyzing momentum simpler, as its effects are clear and calculable using straightforward formulas.

pressure and stress components

The pressure and stress components are encapsulated in the tensor components \(T^{ij}\). These components describe the internal forces experienced by materials in a system or how an object faces external forces. Key points about pressure and stress in Newtonian systems include:

• The magnitude of these components \(\left|T^{ij}\right|\) is far less than both energy density and momentum density.
• They are associated with internal interactions that are small-scale compared to bulk mass energy (e.g., gas pressure in a tank).
• Stress components become more relevant in materials science, where understanding internal forces determines the integrity and deformation of materials.

Due to the relatively small values of pressure and stress in Newtonian scenarios, they have less influence on the system's overall energy balance but are important for material properties and engineering assessments.