General Relativity Lecture 10

TL;DR

In the weak-field approximation, Einstein's vacuum field equations reduce to wave equations for gravitational waves. You write the metric as flat spacetime plus a small correction h, then discard every term quadratic in h. What survives is built purely from second derivatives of h, and equations of that form are wave equations.

Transcript

Stanford University so the questions that were asked to me tonight are more or less by accident they exactly the ones that I want to address tonight weak gravitational fields linearity versus nonlinearity um and gravitational waves let's again working out the equations of general relativity is always unpleasant and we're not going to do it on the B... Read More

Key Insights

• Einstein's vacuum field equations are R_mu_nu minus one-half g_mu_nu R equals zero. Taking the trace of both sides shows the scalar curvature R equals zero, which lets you drop the second term and simplifies the equation.
• The weak-field approximation means gravitational wave amplitudes are small enough that the amplitude squared is treated as zero. When expanding about a small quantity, the rule is to ignore anything higher order in that quantity.
• The only equilibrium solution with no matter and no time dependence is empty flat space with no curvature. Flat spacetime admits coordinates where the metric takes the simple form eta_mu_nu, and it solves the field equations trivially.
• A weak gravitational field is written as the true metric equals eta_mu_nu plus a small correction h_mu_nu, where h is named for the letter after G. Unlike eta, h is a function of position and time and can describe a wave.
• Far from a strong source like a rotating binary pulsar, the gravitational radiation becomes very weak. Even if the field is strong near the source, going far enough away makes the produced waves small enough for the approximation.
• The Christoffel symbol contains the inverse metric times a derivative of the metric. Since derivatives of the constant eta symbol are zero, in this approximation the Christoffel symbol is proportional to derivatives of h.
• The Ricci tensor contains a derivative of the Christoffel symbol plus Christoffel-times-Christoffel. The first gives second derivatives of h, while the quadratic term is much smaller and is ignored, so Ricci is built from second derivatives of h.
• The resulting equation of motion is composed of second derivatives of h with respect to position and time. Equations of that form, especially in relativity, are usually wave equations, which is how gravity waves emerge.

Questions & Answers

Q: How do Einstein's equations become wave equations?

In the weak-field approximation you write the metric as flat spacetime plus a small correction h, then compute the Ricci tensor and set it to zero. After discarding every term that is quadratic in h, what remains is composed entirely of second derivatives of h with respect to position and time. Equations built from second derivatives of that form are, especially in relativity, usually wave equations, and their solutions create the theory of gravity waves.

Q: What does 'weak gravitational waves' mean?

Weak gravitational waves means the amplitudes of the waves are small enough that you can make approximations such as treating the amplitude squared of the gravitational field as zero. When a quantity is small and you expand an equation about that smallness, the usual rule is to ignore terms that are higher order in the small quantity. So squares and products of the small field and its derivatives are dropped as too small to matter.

Q: What is Einstein's field equation without matter?

Without any matter or energy momentum tensor on the right hand side, the equation of motion is R_mu_nu minus one-half g_mu_nu R equals zero, called the Einstein tensor set to zero. If you take the trace of both sides you discover the scalar curvature R equals zero. Once you know that, you can ignore the second term, leaving a simpler equation that describes gravity in a context with no energy momentum tensor.

Q: Why is empty flat space the equilibrium solution?

An equilibrium situation means a solution with no time dependence and no matter, meaning no energy momentum tensor on the right hand side. With no right hand side there is really only one such situation: empty flat space, which is time independent and has no curvature and no interesting gravitational field. Because the Einstein field equations demand certain curvature components vanish, a metric with no curvature at all is trivially a solution.

Q: What is the metric of flat spacetime?

Flat spacetime with no gravitational field admits a choice of coordinates in which the metric takes the simple form eta_mu_nu. It is a matrix with one time component and three space components, where the rows and columns correspond to t, x, y, and z. It is an equilibrium solution of the Einstein field equations because it has no curvature, so plugging it in gives a solution in one trivial step.

Q: What is h_mu_nu in weak field gravity?

When you go far from a source, the true metric can be chosen equal to eta_mu_nu plus something small, and that small correction is called h_mu_nu. It is named h because h is the letter after g. Unlike eta_mu_nu, h_mu_nu is in general a function of position and time; it varies from place to place and can describe a wave. Its components are much smaller than those of the flat metric.

Q: Why can quadratic terms in h be ignored?

Both h and its derivative are treated as small, since differentiating a small smooth function keeps it small. A term like eta times h is once-small, but h times a derivative of h is twice-small, meaning quadratic in the fluctuation. For example, if h is about 0.1, then h times h is about 0.01, far smaller. So products such as Christoffel times Christoffel are dropped, leaving only the linear contributions.

Q: Why does the derivative of the flat metric vanish?

The flat metric eta contains only zeros and ones, which are constants. The derivative of a constant is zero, so the derivative of eta is just zero. Because of this, when you take the derivative of the full metric in the weak-field approximation, only the derivatives of h survive. This is why the Christoffel symbol reduces to being proportional to derivatives of h rather than derivatives of the full metric.

Summary

In this video, the lecturer discusses weak gravitational fields, linearity versus non-linearity, and gravitational waves. They start by explaining that although working out the equations of general relativity is complicated and not suitable for the blackboard, the principles are straightforward. The lecturer goes on to describe weak gravitational waves as small amplitude fluctuations in the gravitational field, and explains how to make approximations when solving for these waves. They then discuss equilibrium situations and how the metric of flat space can be chosen and represented in different coordinates. Moving on to the equations of motion, the lecturer presents a schematic view of the equations, explaining that they have a relatively simple form and resemble wave equations. Finally, they discuss the nature of gravitational waves, the transversality of the fields, and the effect of these waves on the metric tensor.

Questions & Answers

Q: How can the equations of general relativity be summarized?

The equations of general relativity are typically unpleasant to work with on the blackboard, but they can be summarized by the principles and solutions obtained from solving the equations.

Q: What are weak gravitational waves?

Weak gravitational waves refer to small amplitude fluctuations in the gravitational field. They are characterized by the amplitudes of the waves being small enough to make approximations.

Q: Explain the concept of equilibrium situations.

Equilibrium situations describe solutions with no time dependence and no matter on the right-hand side of the equations. In other words, it is a situation of empty space with no curvature or interesting gravitational fields.

Q: How does the metric of flat space depend on coordinates?

The metric of flat space depends on the coordinates used. While one commonly used metric is the Kronecker Delta, there can be other metrics depending on the choice of coordinates. The special feature of flat space is that there are coordinates in which the metric has a simple form.

Q: What are the components of the metric tensor of flat space?

The metric tensor of flat space can be written as a matrix of components: 1 -1 -1 -1 for the first row and column, followed by 0 0 0 0, 0 0 1 0, and 0 0 0 1.

Q: What is the significance of weak gravitational waves being small?

The smallness of weak gravitational waves allows for approximations to be made, such as ignoring higher order terms. This simplifies the equations and makes them more manageable to work with.

Q: How can the equations of motion for weak gravitational waves be summarized?

The equations of motion for weak gravitational waves have a relatively simple form and resemble wave equations. They can be written using second derivatives of the metric tensor and involve the Christoffel symbol, Ricci tensor, and more.

Q: How does the metric tensor change with gravitational waves?

The metric tensor of flat space is modified by gravitational waves, represented by a small correction term called H_mu_nu. This term depends on the position and time coordinates, and describes the wave field.

Q: What are the constraints on gravitational waves derived from Einstein's field equations?

The constraints on gravitational waves require the transversality of the fields, meaning that the time and Z components of H must be zero. Additionally, the trace of H_ij must be zero.

Q: Can you provide an example of the components of the metric tensor for a gravitational wave?

For gravitational waves propagating along the Z axis, the components of the metric tensor that are allowed to be nonzero are H_ij times sine(KX) times sine(KZ-T), where K represents the wave number. The other components are set to zero.

Takeaways

Gravitational waves are small amplitude fluctuations in the gravitational field. They can be approximated as weak gravitational waves, allowing for simplifications in the equations of motion. The metric tensor of flat space can be modified by gravitational waves, characterized by a small correction term called H_mu_nu. The constraints on gravitational waves include transversality of the fields and the trace of H_ij being zero. Gravitational waves can cause tidal forces and deformations in physical objects, making them an interesting field of study.

Summary & Key Takeaways

• Susskind addresses weak gravitational fields, linearity versus nonlinearity, and gravitational waves. Rather than deriving the full equations on the blackboard, he states the principles and the solutions. Weak waves have amplitudes small enough that amplitude-squared terms can be treated as zero when expanding about that smallness.

• He writes Einstein's vacuum equations, R_mu_nu minus one-half g_mu_nu R equals zero, and shows the trace forces R to zero. The only matterless, time-independent equilibrium is empty flat space, whose metric can be chosen as eta_mu_nu with one time and three space components and no curvature.

• A weak field is the flat metric plus a small correction h_mu_nu, a function of position and time. Computing the Ricci tensor and dropping all terms quadratic in h leaves an equation built from second derivatives of h. Such equations are wave equations, producing the theory of gravity waves.