Questions & explanations
1. What is the Gibbons-Hawking-York boundary term and why is it needed?
The Gibbons-Hawking-York (GHY) term is an extra term added to the Einstein-Hilbert action when the spacetime region has a boundary. The usual action's variation gives boundary terms involving derivatives of the metric variation. To have a well-defined variational principle where the metric is fixed on the boundary, those boundary terms must cancel. The GHY term is chosen precisely to cancel them when the variation of the induced metric on the boundary is fixed. It involves the trace of the extrinsic curvature of the boundary. Without it, the action principle would not yield Einstein's equations correctly for bounded regions. The GHY term is essential for path integral formulations and for calculations in asymptotically flat spacetimes.
2. Compare Maxwell's equations in flat spacetime and curved spacetime. What is similar? What is new?
In both flat and curved spacetime, the source-free equations can be written in the same covariant form: dF=0 and d*F=0 (using exterior derivative). The difference is that in curved spacetime, the derivative is covariant, which includes Christoffel symbols. In flat spacetime, these symbols vanish, so the equations become simpler partial derivatives. The new feature in curved spacetime is that the metric appears explicitly in the equations through the covariant derivative. This causes electromagnetic waves to follow curved paths and experience gravitational redshift. Additionally, the wave equation for the potential includes a curvature coupling term. So the equations are structurally similar but have extra terms.
3. Give an example of a pure electric field in one frame appearing as a magnetic field in another.
Take a long line of stationary positive charges. In their rest frame, there is a radial electric field and no magnetic field. If an observer moves parallel to the wire, they see the charges moving, creating a current. That current produces a magnetic field encircling the wire. So a moving observer detects a magnetic field even though the wire is neutral? Actually, careful: In the rest frame of the charges, there is an electric field but no magnetic field. In a frame moving along the wire, the charges are moving, so there is a current and hence a magnetic field. Also, due to length contraction, the charge density changes, resulting in a different electric field. This example shows the relativity of fields.
4. What is the Palatini identity?
The Palatini identity relates the variation of the Ricci tensor to covariant derivatives of variations of the connection. Specifically, the variation of the Ricci tensor equals the difference of two covariant derivatives of the variation of the connection. This identity allows one to vary the connection independently from the metric in the Einstein-Hilbert action. In the Palatini formalism, metric and connection are treated as independent fields. The identity simplifies the derivation of Einstein's equations. It shows that varying the connection gives the condition that the connection is the Levi-Civita one (torsion-free and metric-compatible) when matter does not couple to connection.
5. How does the Palatini variation help derive Einstein's equations from the action?
In the Palatini approach, the Einstein-Hilbert action is varied with respect to both the metric and the connection. The Palatini identity expresses the variation of the Ricci scalar in terms of connection variations. Then, integration by parts gives a surface term and an equation from the connection variation. That equation forces the connection to be the Levi-Civita connection (if no torsion). Substituting back, the metric variation yields the usual Einstein equations. This derivation shows that the metric and connection are not truly independent; the connection is determined by the metric from the equations of motion. It provides a different insight into the variational principle.
6. How does adding the GHY term affect the variational principle?
Without the GHY term, variation of the Einstein-Hilbert action produces boundary terms that involve the normal derivative of the metric variation. To make the action stationary, those boundary terms must vanish, which imposes conditions on the derivative of the metric at the boundary. The GHY term is designed to cancel exactly those boundary terms when the induced metric is fixed. Then the variational principle is well-defined: the action is stationary for solutions of Einstein's equations with the induced metric fixed on the boundary. This is similar to how a surface term is needed in other field theories. It makes the action principle self-consistent.
7. How does the matter content of the universe affect the solutions of the field equations?
The matter content enters the field equations through the stress-energy tensor. Different types of matter have different relations between density and pressure. For ordinary matter (dust), pressure is almost zero. For radiation, pressure is one-third of density. For dark energy, pressure is negative and close to minus density. These differences affect how the universe expands over time. For example, a matter-dominated universe expands slower than a radiation-dominated one. The present universe is dominated by dark energy, which causes accelerated expansion. The field equations show that the expansion history is determined by the mix of these components.
8. Given a specific curved spacetime, say the Schwarzschild metric, how would you write the source-free Maxwell equations in coordinates?
For the Schwarzschild metric, you can compute the covariant derivatives of the field strength. The inhomogeneous equation ∇_μ F^{μν}=0 becomes a set of partial differential equations with metric coefficients. For example, the t-component involves derivatives of the radial and angular components. The homogeneous equation through the dual tensor gives similar equations. One often uses the vector potential A_μ and writes the wave equation □A_μ - R_μ^ν A_ν = 0 in the Lorenz gauge, where □ is the d'Alembertian in curved space. This yields specific solutions like electromagnetic waves bending around the black hole. The equations are coupled and non-trivial.
9. Why is a car speeding up not an inertial frame of reference?
A car speeding up is accelerating, so it is not an inertial frame. In an accelerating car, you feel a force pushing you back into the seat. Objects inside the car, like a cup on the dashboard, will slide backwards unless held. This violates Newton's first law, which says that without a net force, objects should keep their state of motion. In an inertial frame, no such fake forces appear; objects stay still or move uniformly. Because the laws of physics look different when you accelerate – for instance, the cup moves even though no real force acts on it – we cannot apply the principle of relativity. So special relativity only works in inertial frames.
10. Give an everyday example that illustrates the constancy of the speed of light.
Think of two cars: one coming towards you and one going away. If they honk their horns, the sound from the approaching car sounds higher in pitch, and the sound from the receding car sounds lower. That is the Doppler effect for sound. But for light, if a star moves towards or away from Earth, the color shifts in a similar way. However, the speed of the light itself does not change; it always travels at c. Even if a spaceship flies toward Earth at half the speed of light and shines a laser, the laser light still arrives at Earth at speed c, not faster. This constant speed light is different from sound, which does change speed relative to the medium.
11. In flat spacetime, conservation of energy-momentum is just partial derivatives. How does it change in curved spacetime?
In flat spacetime, conservation uses ordinary partial derivatives, so the sum of derivatives of stress-energy components is zero. In curved spacetime, we must use covariant derivatives that account for the changing metric. The contracted Bianchi identity forces the covariant divergence of stress-energy to be zero. This means the conservation law includes extra terms from the connection (Christoffel symbols). The physical idea is that energy and momentum can be transferred to or from the gravitational field, so the matter alone is not conserved with ordinary derivatives. But with covariant derivatives, the total matter plus geometry is conserved.
12. What was the Michelson-Morley experiment trying to measure?
The Michelson-Morley experiment aimed to measure the Earth's motion through a hypothetical substance called aether, which was thought to be needed for light to travel. Scientists believed that the speed of light would change depending on the direction relative to the Earth's movement through the aether. The experiment used a device called an interferometer to compare light speeds in two perpendicular directions. It was expected that the light speed difference would reveal the Earth's speed through the aether. However, the experiment found no difference at all, which surprised everyone. This null result meant that the idea of aether was wrong.