Start Learning Gravity

Start with the work you want to be able to do. This page asks only the preparation questions that matter for that goal, then gives one of three results: ready, review, or repair. It does not ask about degrees, job titles, grades, or how long you have studied. You may always ignore its recommendation and choose any route directly.

This route requires no account or timer, uses no JavaScript, and stores no answers. If you want a private record, copy the short result table into your own notes.

If a copied or bookmarked result names a route that is no longer offered, that result is stale. Treat the named route as unavailable, assign no ready, review, or repair result, and choose again from the routes currently shown here. An unavailable route is neither displayed as a link nor replaced by a guessed recommendation.

Choose the work

Pick the row that best describes the next piece of work—not a permanent label for your background.

I want to…	Begin here	Readiness check
form a first accurate picture of gravity and black holes	First Encounter with Gravity and Black Holes	none; begin directly
prepare the mathematics and physics used in relativity	Mathematics and Physics for Relativity or the preparation directory	none; the path diagnoses each capability as it arises
learn graduate general relativity	Graduate General Relativity Self-Study Path or the First Graduate Course	calculus and linear algebra; differential equations; special relativity; tensor geometry
specialize in a branch of gravity	Learning Paths	use the selected path’s own prerequisite check
prepare for the Evidence and Open Problems seminar	Research Seminar: Evidence and Open Problems in Gravitation or the Research Atlas	seminar common entry only: Einstein’s equation; elementary inference; selected technical cases add prerequisites
assemble a coherent course	Courses and Seminars	compare audience, outcomes, prerequisites, assignments, and capstone

The two goals with a readiness check are worked out below. For every prompt, try the task without notes and mark the result:

• secure — you can complete it and explain why the method works;
• uncertain — the main idea is familiar, but your explanation or execution has a gap; or
• not yet — you cannot presently begin, or would have to imitate steps you could not justify.

Use the least favorable result honestly. A single not yet means repair. With no not yet but at least one uncertain, the result is review. All secure means ready for that goal’s stated entry tasks. Use this result to choose what to study next.

Readiness for graduate general relativity

Graduate relativity uses four capabilities immediately and repeatedly. Work the four checks in order. Each one is deliberately small: it tests whether a tool is available, not whether you have already learned general relativity.

1. Calculus and linear algebra

Let

$$A=\begin{pmatrix}2&1\\1&2\end{pmatrix}, \qquad v=\begin{pmatrix}1\\-1\end{pmatrix}.$$

Find $Av$, identify the eigenvalue belonging to $v$, and explain in one sentence why the eigenvalue is independent of multiplying $v$ by a nonzero constant. Then differentiate $f(x,y)=x^2e^y$ with respect to both variables.

Check the reasoning

$Av=(1,-1)^T=v$, so the eigenvalue is $1$. If $Av=\lambda v$, then $A(cv)=cAv=\lambda(cv)$ for every nonzero $c$; rescaling changes the representative vector, not the eigendirection or eigenvalue. The derivatives are $\partial_x f=2xe^y$ and $\partial_y f=x^2e^y$.

Mark secure only if both the calculation and rescaling explanation are clear. If the calculation is right but the explanation is shaky, mark uncertain.

For review or repair, use Calculus and Linear Algebra for Relativity.

2. Differential equations

For $\omega\gt0$, solve

$$y''+\omega^2y=0, \qquad y(0)=a, \qquad y'(0)=0,$$

and name a quantity built from $y$ and $y'$ that remains constant.

Check the reasoning

The solution is $y(x)=a\cos(\omega x)$. One conserved quantity is $E=\tfrac12[(y')^2+\omega^2y^2]$; differentiating gives $E'=y'(y''+\omega^2y)=0$. A correct formula without the substitution or conservation check is evidence for uncertain, not secure.

For review or repair, use Differential Equations and Qualitative Dynamics.

3. Special relativity

In units with $c=1$, two events have coordinate separation $\Delta t=5$ and $\Delta x=3$ in an inertial frame. Classify the separation, compute the proper-time interval using the site’s $(-,+,+,+)$ convention, and say which part of your answer every inertial observer must agree on.

Check the reasoning

$\Delta s^2=-\Delta t^2+\Delta x^2=-16$, so the separation is timelike and $\Delta\tau=\sqrt{-\Delta s^2}=4$. Other inertial observers assign different coordinate differences, but agree on the interval, its timelike character, and the proper time between the events. If signs were guessed rather than connected to the declared convention, mark uncertain.

For review or repair, use Mechanics and Special Relativity Diagnostic.

4. Tensor geometry

Explain why the components of a vector change under a coordinate transformation while the geometric vector does not. Then say why Christoffel symbols can vanish at a point in a suitable coordinate system even when the Riemann curvature there is nonzero.

Check the reasoning

A vector is a coordinate-independent tangent-space element; its components are coefficients in a chosen basis, and the basis changes with coordinates. Christoffel symbols are connection coefficients rather than tensor components, so their inhomogeneous transformation law permits them to vanish at one point. Curvature is tensorial and depends on derivatives and quadratic combinations of the connection, so nonzero curvature cannot be transformed away. Mark secure only if both distinctions are explicit.

For review or repair, use Tensor Geometry Bridge.

Route the result

Result	Next action
ready	Begin the Graduate General Relativity Self-Study Path. Choose the First Graduate Course instead when you want fixed assignments and assessment criteria.
review	Open the linked module for every uncertain capability, attempt its diagnostic and transfer problem, then return to the graduate path.
repair	Begin with the first linked module marked not yet. Complete its review sequence and changed-setting transfer problem before checking the next gap. Then return to the graduate path.

Multiple gaps do not require completing every preparation page. Review the relevant topics in the order above because later checks use earlier tools. If a module’s opening diagnostic is already secure, take its transfer problem and move on.

Readiness for the Evidence and Open Problems seminar

This is the common entry diagnostic for one seminar, not a general test of research readiness. It checks whether you can recover one classical-gravity argument and interpret elementary Bayesian evidence language used across the seminar’s cases. Passing it does not establish readiness for research across gravity, nor does it supply the technical preparation for every case.

Before a bounded technical reproduction, the seminar applies the selected case’s own prerequisite profile. Advanced learning paths likewise own their branch-specific entry checks; readiness for one branch must not be inferred from the two common tasks below.

1. Einstein’s equation

Write Einstein’s equation with a cosmological constant. Identify its geometric and matter sides, and explain why covariant conservation of stress-energy is consistent with the equation.

Check the reasoning

With the site’s convention,

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}.$$

The left side is geometric and the right side represents matter and non-gravitational fields. The contracted Bianchi identity gives $\nabla^\mu G_{\mu\nu}=0$, while metric compatibility gives $\nabla^\mu g_{\mu\nu}=0$; for constant $\Lambda$, the equation therefore implies $\nabla^\mu T_{\mu\nu}=0$. Mark uncertain if you can state the equation but cannot supply that chain of reasoning.

For review or repair, study Einstein Field Equations and check each convention before carrying a formula into a seminar case.

2. Probability and inference

Suppose a model has parameter $\theta$ and data $d$. State the roles of the likelihood, prior, posterior, and evidence in Bayes’ theorem. Then explain in one sentence why an unmodeled selection effect can bias an inferred population.

Check the reasoning

Bayes’ theorem reads

$$p(\theta\mid d)=\frac{p(d\mid\theta)p(\theta)}{p(d)}.$$

$p(d\mid\theta)$ is the likelihood, $p(\theta)$ the prior, $p(\theta\mid d)$ the posterior, and $p(d)$ the marginal likelihood or evidence. A selection process changes which systems enter the observed sample; if that detection probability is omitted, the observed distribution is mistaken for the underlying population. Mark secure only if the symbols and the selection argument are both interpretable in words.

For review or repair, use Probability and Inference for Gravity.

Route the result

All secure means that the seminar’s two common entry tasks are available; it leads to Research Seminar: Evidence and Open Problems in Gravitation, not to a claim of domain-general research readiness. Any uncertain result calls for the linked review before the seminar; any not-yet result calls for repair. The selected case still applies its own prerequisite check before technical reproduction. If your goal is to inspect dated evidence without undertaking the seminar assignments, browse the Research Atlas directly.

Override, stop, or return

A diagnostic can be wrong about what you need. You may know a tool in a notation unlike the prompt, have an accessibility reason to demonstrate it in another form, or be deliberately exploring ahead of your preparation. Choose the direct route whenever that serves the work, and use its first checkpoint as better evidence.

If you stop, retain only three facts: the goal you chose, the least secure capability, and the linked next page. On returning, repeat that capability’s small check. The route is designed to reconstruct its state from your work, without a browser history or saved profile.