A Consistency Test for Kerr Black Holes via Orbital Motion, Ringdown, and Imaging

Posted on January 11, 2026 by Aronno Mirdha

Kerr Trisector Closure (KTC) is a consistency test for the Kerr hypothesis that tries to stay honest about what is actually being inferred from data. The guiding principle is simple: if the exterior spacetime of an astrophysical, stationary, uncharged black hole is Kerr, then there exist parameters $(M,\chi)$ such that every observable in every channel is generated by the same Kerr geometry with those parameters. KTC is just the exercise of turning that sentence into a clean mathematical statement that you can put into a likelihood analysis.

Start with the Kerr family written abstractly as a two-parameter set of spacetimes

$$
\mathcal{K}={( \mathcal{M}, g_{ab}(M,\chi) ) : M>0,; |\chi|<1},
$$

where $M$ is the mass and $\chi = J/M^2$ is the dimensionless spin (using geometric units $G=c=1$). The only thing we will use about Kerr is that any prediction for any measurement in a given “sector” is a deterministic functional of $(M,\chi)$ once you fix nuisance choices like orientation, distance, inclination, etc.

Now define three observational sectors:

Orbital sector $\mathsf{O}$: timelike dynamics, e.g. orbital frequencies, precessions, inspiral phasing in the adiabatic regime.

Ringdown sector $\mathsf{R}$: quasi-normal mode (QNM) spectrum, i.e. complex frequencies $\omega_{\ell m n}(M,\chi)$ and associated amplitudes/phases.

Imaging sector $\mathsf{I}$: null geodesics and radiative transfer, e.g. shadow size/asymmetry, photon ring structure, closure phases, visibility amplitudes.

For each sector $s \in {\mathsf{O},\mathsf{R},\mathsf{I}}$, let $d_s$ denote the data in that sector. A standard statistical model is: conditional on parameters, the data are distributed according to a likelihood

$$
\mathcal{L}_s(d_s \mid \theta_s, \lambda_s),
$$

where

$$
\theta_s = (M_s,\chi_s)
$$

are the Kerr parameters inferred from that sector, and $\lambda_s$ collects nuisance parameters for that sector (distance, inclination, calibration, environment, waveform systematics, scattering, emissivity model, etc.). The key point is not what is inside $\lambda_s$ but that the likelihood for each sector can be written down, at least in principle, as a function of $(M,\chi)$ plus nuisances.

At this point, you have two different hypotheses you can formalize.
1. The “unconstrained” model says each sector can have its own Kerr parameters:

$$
H_{\text{free}}:\quad \theta_{\mathsf{O}},\theta_{\mathsf{R}},\theta_{\mathsf{I}} \text{ are independent a priori.}
$$

  1. The “closure” model says there is a single Kerr spacetime behind all three:

$$
H_{\text{Kerr}}:\quad \theta_{\mathsf{O}}=\theta_{\mathsf{R}}=\theta_{\mathsf{I}}=\bar\theta,
$$

for some common $\bar\theta = (\bar M,\bar\chi)$.

KTC is the act of comparing these two, or equivalently quantifying how strongly the data prefer a shared $(M,\chi)$ over three separate ones.

To make this rigorous, write the evidence (marginal likelihood) under each model. Under $H_{\text{Kerr}}$, the joint likelihood factorizes across sectors conditional on the shared parameters (this is the usual conditional-independence assumption given the source parameters; if you have shared systematics you can explicitly couple them in the nuisance structure):
$$
\mathcal{L}(d_{\mathsf{O}},d_{\mathsf{R}},d_{\mathsf{I}}\mid \bar\theta,\bar\lambda)
=\prod_{s\in{\mathsf{O},\mathsf{R},\mathsf{I}}} \mathcal{L}s(d_s\mid \bar\theta,\lambda_s),
$$
with $\bar\lambda = (\lambda{\mathsf{O}},\lambda_{\mathsf{R}},\lambda_{\mathsf{I}})$. The evidence is then
$$
Z_{\text{Kerr}}
=\int \left[\prod_{s}\mathcal{L}_s(d_s\mid \bar\theta,\lambda_s)\right],
\pi(\bar\theta),\prod_s \pi(\lambda_s), d\bar\theta, d\lambda_s.
$$

Under $H_{\text{free}}$, you allow independent Kerr parameters per sector:
$$
Z_{\text{free}}
=\int \left[\prod_{s}\mathcal{L}_s(d_s\mid \theta_s,\lambda_s)\right],
\left[\prod_s \pi(\theta_s)\pi(\lambda_s)\right],
\prod_s d\theta_s, d\lambda_s.
$$

A clean closure statistic is the Bayes factor
$$
\mathcal{B}=\frac{Z_{\text{Kerr}}}{Z_{\text{free}}}.
$$
If $\mathcal{B}$ is large, the data prefer the shared-parameter Kerr description. If $\mathcal{B}$ is small, the data prefer letting the sectors drift apart in $(M,\chi)$, which is exactly what “failure of closure” means in a statistically coherent way.

If you prefer a frequentist formulation, you can do essentially the same thing with a constrained versus unconstrained maximum-likelihood comparison. Define the log-likelihoods
$$
\ell_{\text{free}} = \sum_s \max_{\theta_s,\lambda_s} \log \mathcal{L}s(d_s\mid \theta_s,\lambda_s),
$$
$$
\ell{\text{Kerr}} = \max_{\bar\theta,\bar\lambda} \sum_s \log \mathcal{L}s(d_s\mid \bar\theta,\lambda_s).
$$
Then a likelihood-ratio test statistic is
$$
\Lambda = 2(\ell{\text{free}}-\ell_{\text{Kerr}}).
$$
Heuristically, $\Lambda$ measures the “penalty” for forcing the three sectors to share the same $(M,\chi)$. Under regularity conditions and in an asymptotic regime, $\Lambda$ is approximately $\chi^2$ distributed with degrees of freedom equal to the number of constraints, which here is $4$ (two parameters per sector, three sectors gives $6$ parameters, constrained model has $2$). In practice, because the models can be nonlinear and posteriors non-Gaussian, you calibrate $\Lambda$ by simulation.

So far, this is structure and not physics. The physics enters when you specify what each sector is actually measuring, meaning how $(M,\chi)$ shows up in observables.

For the orbital sector, the typical statement is that certain gauge-invariant frequencies (azimuthal, radial, polar) for bound Kerr geodesics are functions of $(M,\chi)$ and constants of motion. A standard representation is
$$
\Omega_i = \Omega_i(M,\chi; p,e,\iota),
$$
where $(p,e,\iota)$ parametrize the orbit (semi-latus rectum, eccentricity, inclination), and $i$ ranges over the fundamental frequencies. Observationally you do not measure $(p,e,\iota)$ directly; they become part of the nuisance structure or dynamical parameterization, but the core point remains: the orbital likelihood has a map from $(M,\chi)$ into predicted timing and phasing data.

For the ringdown sector, the measurable quantities are complex mode frequencies. For a Kerr black hole,
$$
\omega_{\ell m n} = \frac{1}{M}, f_{\ell m n}(\chi),
$$
for some dimensionless functions $f_{\ell m n}$ determined by black hole perturbation theory (Teukolsky equation with appropriate boundary conditions). The $1/M$ scaling is exact because Kerr has no length scale other than $M$ in geometric units, and the dependence on $\chi$ is encoded in the dimensionless eigenvalue problem. The ringdown likelihood is built from comparing measured $(\Re \omega, \Im \omega)$ (and amplitudes) to these predictions.

For the imaging sector, the clean geometric object is the photon region and its projection onto the observer sky. The boundary of the Kerr shadow can be written in terms of critical impact parameters that are functions of $(M,\chi)$ and the observer inclination $i$. One convenient parameterization uses the constants of motion $(\xi,\eta)$ for null geodesics and gives celestial coordinates $(\alpha,\beta)$ on the image plane:
$$
\alpha = -\frac{\xi}{\sin i}, \qquad
\beta = \pm \sqrt{\eta + a^2\cos^2 i – \xi^2 \cot^2 i},
$$
with $a=\chi M$. The critical curve is obtained by selecting $(\xi,\eta)$ corresponding to unstable spherical photon orbits. Again, the details are not the point here; the point is that there is an explicit deterministic forward model from $(M,\chi)$ (plus inclination and emission model nuisances) to interferometric observables.

Once you accept that each sector admits a forward model, the closure logic becomes almost tautological: Kerr predicts that the same $(M,\chi)$ must fit all three forward models simultaneously. KTC is the quantitative version of “simultaneously.”

It is also useful to write the closure condition in a way that looks like something you can plot. From each sector you can compute a posterior for $\theta_s$ after marginalizing nuisance parameters:
$$
p_s(\theta_s\mid d_s)
\propto \int \mathcal{L}s(d_s\mid \theta_s,\lambda_s),\pi(\theta_s),\pi(\lambda_s), d\lambda_s.
$$
Under the closure hypothesis, you want these to be consistent with a single $\bar\theta$. A natural diagnostic is the product posterior (sometimes called posterior pooling under conditional independence)
$$
p{\text{pool}}(\bar\theta \mid d_{\mathsf{O}},d_{\mathsf{R}},d_{\mathsf{I}})
\propto \pi(\bar\theta)\prod_s \frac{p_s(\bar\theta\mid d_s)}{\pi(\bar\theta)},
$$
which simplifies to a product of likelihood contributions if the priors are aligned. In a Gaussian approximation where each sector yields a mean $\mu_s$ and covariance $\Sigma_s$ in the $(M,\chi)$ plane, you can make the closure tension extremely explicit. The pooled estimator has covariance
$$
\Sigma_{\text{pool}}^{-1} = \sum_s \Sigma_s^{-1},
$$
and mean
$$
\mu_{\text{pool}} = \Sigma_{\text{pool}}\left(\sum_s \Sigma_s^{-1}\mu_s\right).
$$
Then a standard quadratic tension statistic is
$$
T = \sum_s (\mu_s-\mu_{\text{pool}})^{\mathsf{T}}\Sigma_s^{-1}(\mu_s-\mu_{\text{pool}}).
$$
If Kerr is correct and the error models are accurate, $T$ should be “reasonable” relative to its expected distribution (again, best calibrated by injection studies). If $T$ is systematically large, you have a closure failure.

What is nice about this formulation is that it cleanly separates two questions that often get mixed up.

First: does each sector individually admit a Kerr fit? This is about whether $\mathcal{L}_s$ has support near some Kerr parameters.

Second: are the Kerr parameters inferred by each sector consistent with each other? This is the closure question. You can pass the first and fail the second, and that second failure is exactly the kind of thing you would expect from a theory that mimics Kerr in one channel but not in another, or from unmodeled environmental/systematic effects that contaminate one sector differently.

So the rigorous derivation of KTC is basically the derivation of a constrained inference problem. You define three sector likelihoods with their own nuisances, you impose the identification $\theta_{\mathsf{O}}=\theta_{\mathsf{R}}=\theta_{\mathsf{I}}$, and you quantify the loss of fit relative to the unconstrained model. Everything else is implementation detail.

If you want one line that captures the whole test, it is this: KTC is the comparison of
$$
H_{\text{Kerr}}:\exists,\bar\theta\text{ such that }\forall s,; d_s \sim \mathcal{L}s(\cdot\mid \bar\theta,\lambda_s)
$$
against
$$
H{\text{free}}:\forall s,\exists,\theta_s\text{ such that } d_s \sim \mathcal{L}_s(\cdot\mid \theta_s,\lambda_s),
$$
with the comparison done via a Bayes factor $\mathcal{B}$ or a likelihood-ratio statistic $\Lambda$.

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Towards a derivation of the metric tensor in general relativity

Posted on January 6, 2026 by Aronno Mirdha

One of the central tasks in differential geometry is to make precise the notion of length and angle on a smooth manifold. Unlike $\mathbb R^n$, a general manifold comes with no preferred inner product. The metric tensor is not something that “appears” automatically; rather, it is a geometric structure we deliberately introduce. The goal of this post is to derive the metric tensor from first principles in a clean, logically economical way.

We begin with a smooth $n$-dimensional manifold $M$. At each point $p\in M$, we want to measure lengths of infinitesimal displacements through $p$. Infinitesimal displacements are modeled by tangent vectors, so the correct object to define is an inner product on each tangent space $T_pM$.

A precise and coordinate-free definition of tangent vectors is the following. A tangent vector at $p$ is a derivation at $p$: a linear map $X:C^\infty(M)\to\mathbb R$ satisfying the Leibniz rule $X(fg)=f(p)X(g)+g(p)X(f)$. The collection of all such derivations forms a real vector space $T_pM$ of dimension $n$.

Now introduce a coordinate chart $(U,\varphi)$ with $\varphi=(x^1,\dots,x^n)$ and $p\in U$. For each coordinate function, define a derivation by

$$ \left.\frac{\partial}{\partial x^i}\right|_p (f) := \frac{\partial (f\circ\varphi^{-1})}{\partial u^i}\Big|_{\varphi(p)}. $$

These derivations form a basis of $T_pM$. Every tangent vector $X\in T_pM$ can therefore be written uniquely as

$$ X = X^i \left.\frac{\partial}{\partial x^i}\right|_p $$

for real coefficients $X^i$. These coefficients depend on the chosen coordinates, but the vector $X$ itself does not. At this point, nothing like length exists yet. To measure length, we must specify an inner product on each tangent space.

A Riemannian metric on $M$ is a rule that assigns to each point $p\in M$ a bilinear map

$$ g_p : T_pM \times T_pM \to \mathbb R $$

such that:

  1. $g_p$ is symmetric: $g_p(X,Y)=g_p(Y,X)$
  2. $g_p$ is positive definite: $g_p(X,X)>0$ for all $X\neq 0$
  3. The assignment $p\mapsto g_p(X_p,Y_p)$ is smooth whenever $X,Y$ are smooth vector fields

This definition is entirely coordinate-free. The metric is simply a smoothly varying inner product on tangent spaces.

Now fix a coordinate chart $(x^1,\dots,x^n)$. Since $g_p$ is bilinear, it is completely determined by its values on basis vectors. Define

$$ g_{ij}(p) := g_p\!\left(\left.\frac{\partial}{\partial x^i}\right|_p,\left.\frac{\partial}{\partial x^j}\right|_p\right). $$

These functions $g_{ij}$ are smooth, symmetric in $i,j$, and vary from point to point. They are the components of the metric tensor in coordinates.

Given two tangent vectors

$$ X = X^i \frac{\partial}{\partial x^i}, \qquad Y = Y^j \frac{\partial}{\partial x^j}, $$

bilinearity immediately gives

$$ g_p(X,Y) = g_{ij}(p)\, X^i Y^j. $$

This is the familiar coordinate expression of the metric. Importantly, this is not a definition but a representation of the underlying geometric object $g$.

From this formula, the squared length of a tangent vector $X$ is

$$ |X|^2 = g_{ij}(p)\,X^i X^j. $$

Thus, the metric tensor generalizes the Euclidean dot product by allowing the coefficients $g_{ij}$ to vary with position and coordinates.

It is instructive to check how these components transform. If we change coordinates from $x^i$ to $\tilde x^a$, then the basis vectors transform via the chain rule:

$$ \frac{\partial}{\partial \tilde x^a} = \frac{\partial x^i}{\partial \tilde x^a}\frac{\partial}{\partial x^i}. $$

Substituting into the definition of the metric components yields

$$ \tilde g_{ab} = g_{ij}\frac{\partial x^i}{\partial \tilde x^a}\frac{\partial x^j}{\partial \tilde x^b}. $$

This transformation law is exactly what characterizes $g_{ij}$ as the components of a $(0,2)$-tensor field. The metric tensor is therefore not merely a matrix-valued function but a genuine tensorial object on the manifold.

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Overdetermined parameter interference in physics

Posted on December 31, 2025 by Aronno Mirdha

In many areas of physics, a system is described by a small number of fundamental parameters, while the available observations greatly exceed this number. When this occurs, the problem of parameter inference becomes overdetermined. Rather than being a drawback, this redundancy often plays a central role in testing the internal consistency of a physical theory. A simple example arises when a system depends on a parameter vector $\Theta$ with only a few components, while multiple observables depend on $\Theta$ in different ways. Each observable provides an estimate of the same underlying parameters, but with its own uncertainty and systematic effects. In an idealized setting, these estimates should agree up to statistical noise.

More concretely, suppose a theory predicts that several quantities $\mathcal{O}_k$ depend on a common parameter $\Theta$,
$$
\mathcal{O}_k = \mathcal{O}_k(\Theta).
$$
Measurements of the observables then yield a collection of inferred parameter values $\hat{\Theta}_k$. If the theory is correct and the modeling assumptions are adequate, these inferred values should cluster around a single underlying parameter point.

This situation is familiar in classical mechanics, where the mass of an object can be inferred from its response to different forces, or in electromagnetism, where charge can be inferred from both static and dynamical measurements. In such cases, agreement between independent inferences provides confidence that the underlying description is self-consistent.

In more complex settings, overdetermination becomes a diagnostic tool rather than a mere redundancy. Discrepancies between inferred parameters can signal unmodeled effects, underestimated uncertainties, or a breakdown of the theoretical assumptions used to relate observables to parameters. Importantly, this type of test does not require proposing an alternative theory. It only checks whether a single theoretical framework can simultaneously account for multiple manifestations of the same system.

From a statistical perspective, overdetermined inference naturally leads to goodness-of-fit tests. One seeks a single parameter value $\bar{\Theta}$ that best reconciles all measurements, and then asks whether the residual discrepancies are consistent with the stated uncertainties. Failure of such a reconciliation indicates tension between different pieces of data, even if each measurement individually appears reasonable.

In gravitational physics, overdetermination is particularly natural. A spacetime geometry governs a wide range of physical phenomena, including particle motion, wave propagation, and light deflection. If these phenomena are all described by the same metric, then independent observations should converge on the same geometric parameters. The more distinct the physical processes involved, the more stringent the resulting consistency requirement becomes.

The broader lesson is that overdetermination is not merely a technical feature of data analysis. It reflects a structural property of physical theories that describe systems through a small set of fundamental parameters. When many different observables depend on the same parameters, consistency across these observables becomes a powerful and largely model-independent test of the theory itself.

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A Long-Term Narrowband Astrophotography Project

Posted on December 27, 2025 by Aronno Mirdha

This project documents a long-term narrowband astrophotography dataset accumulated over roughly two years and more than thirty imaging sessions, with a total integration time approaching two hundred hours. Data were acquired using a reduced refracting optical system and an SHO narrowband filter set, then calibrated, stacked, and mosaicked to produce a deep composite image. The result reflects incremental data accumulation across multiple observing seasons rather than a single planned campaign. Continue reading →

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The Kerr Trisector Closure: A project on internal consistency tests of General Relativity

Posted on December 20, 2025 by Aronno Mirdha

General Relativity describes gravity as the curvature of spacetime. Mass and energy determine this curvature, and physical phenomena such as orbital motion, gravitational radiation, and the propagation of light are governed by the resulting geometry. In this framework gravity is not a force but a geometric property of spacetime itself.

Over the past century, General Relativity has been tested in many independent regimes. Planetary motion agrees with relativistic predictions, gravitational waves from compact binary mergers have been observed, and direct images of black hole shadows have been produced. Each of these observations probes a different physical manifestation of spacetime curvature. However, these tests are typically analyzed in isolation. Orbital dynamics, gravitational-wave signals, and black hole imaging are treated as separate probes, even when they concern the same astrophysical object. As a result, current tests do not directly verify whether all observations of a single black hole are mutually consistent with one and the same spacetime geometry.

The aim of this project is to address this missing consistency check. The guiding question is whether independent observations of the same black hole all describe the same spacetime, as predicted by General Relativity.

An uncharged, rotating black hole is described in General Relativity by the Kerr solution. This solution specifies the spacetime geometry outside the black hole and depends on only two physical parameters: the mass $M$ and the dimensionless spin $\chi$. The mass sets the overall curvature scale, while the spin controls the rotation and associated frame-dragging effects. The dimensionless spin is defined by

$$
\chi = \frac{J}{M^2}, \qquad |\chi| \le 1.
$$

A black hole cannot be observed directly. Instead, its spacetime geometry is inferred through its influence on matter, radiation, and light. In this work we focus on three observational sectors, each sensitive to different physical processes but governed by the same underlying Kerr spacetime.

In the dynamical sector, the curvature of spacetime determines the motion of massive bodies. In compact binary systems this motion produces gravitational waves whose phase evolution depends on the mass and spin of the black hole. Analysis of the inspiral signal yields an estimate $(M_{\mathrm{dyn}}, \chi_{\mathrm{dyn}})$.

In the ringdown sector, a perturbed black hole relaxes to equilibrium through a set of damped oscillations known as quasinormal modes. The frequencies and decay times of these modes depend only on the black hole mass and spin, independent of the details of the perturbation. Measurements of the ringdown signal provide a second estimate $(M_{\mathrm{rd}}, \chi_{\mathrm{rd}})$.

In the imaging sector, strong gravitational lensing near the black hole bends light into characteristic structures such as the photon ring and shadow. High-resolution images constrain the spacetime geometry through the size and shape of these features, leading to a third estimate $(M_{\mathrm{img}}, \chi_{\mathrm{img}})$.

The central hypothesis of this project is that if General Relativity correctly describes gravity, then these three independently inferred parameter pairs must be statistically consistent with a single Kerr spacetime. Differences between the measurements are expected due to experimental uncertainty, but they should not exceed what is allowed by those uncertainties. A statistically significant disagreement would indicate that the observations cannot be described by one common spacetime geometry.

To formalize this test, we describe the spacetime by the parameter vector $\Theta = (M, \chi)$. Each observational sector produces an estimate $\hat{\Theta}_{\mathrm{dyn}}$, $\hat{\Theta}_{\mathrm{rd}}$, and $\hat{\Theta}_{\mathrm{img}}$, with corresponding covariance matrices $\Sigma_{\mathrm{dyn}}$, $\Sigma_{\mathrm{rd}}$, and $\Sigma_{\mathrm{img}}$ encoding their uncertainties.

If all sectors are consistent, there should exist a common best-fit parameter vector $\bar{\Theta} = (\bar{M}, \bar{\chi})$ that minimizes the weighted discrepancy

$$
\chi^2(\Theta) = \sum_k
(\hat{\Theta}_k – \Theta)^{\mathrm{T}} \Sigma_k^{-1} (\hat{\Theta}_k – \Theta),
\qquad
k \in \{\mathrm{dyn}, \mathrm{rd}, \mathrm{img}\}.
$$

The minimizing value is given by

$$
\bar{\Theta}
=
\left(\sum_k \Sigma_k^{-1}\right)^{-1}
\left(\sum_k \Sigma_k^{-1} \hat{\Theta}_k\right).
$$

We then define the deviation of each sector from the common spacetime as $\delta \Theta_k = \hat{\Theta}_k – \bar{\Theta}$. The Kerr Trisector Closure statistic is

$$
T^2
=
\sum_k
\delta \Theta_k^{\mathrm{T}} \Sigma_k^{-1} \delta \Theta_k.
$$

This quantity measures the total inconsistency between the three sector estimates while accounting for their uncertainties. Since three independent measurements of two parameters are compared, the statistic follows a $\chi^2$ distribution with four degrees of freedom under the null hypothesis of consistency.

To validate the method, we test it using simulated data. We choose a true spacetime $\Theta^\ast = (M^\ast, \chi^\ast)$ and generate synthetic measurements $\hat{\Theta}_k = \Theta^\ast + \varepsilon_k$, where the errors $\varepsilon_k$ are drawn from Gaussian distributions with covariance $\Sigma_k$. In the consistent case, the resulting $T^2$ values follow the expected $\chi^2_4$ distribution. In Monte Carlo simulations, the mean value and rejection rate match theoretical expectations.

We then introduce a controlled bias in one sector, for example $\hat{\Theta}_{\mathrm{img}} = \Theta^\ast + \Delta + \varepsilon_{\mathrm{img}}$, while leaving the other sectors unchanged. In this case the $T^2$ distribution shifts to larger values and exceeds the consistency threshold with high probability. The individual contributions

$$
T_k^2 = \delta \Theta_k^{\mathrm{T}} \Sigma_k^{-1} \delta \Theta_k
$$

identify the sector responsible for the inconsistency.

These tests show that the Kerr Trisector Closure behaves as intended. It remains statistically quiet when all observations are consistent and responds strongly when a genuine inconsistency is present. The method provides a model-independent way to test the internal consistency of black hole spacetime measurements. Current observations do not yet allow a full experimental implementation, but future gravitational-wave detectors and improved black hole imaging may make such tests possible.

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Spacetime as a Lorentzian Manifold

Posted on November 1, 2025 by Aronno Mirdha

One of the central tasks in differential geometry is to make precise the notion of length and angle on a smooth manifold. Unlike $\mathbb R^n$, a general manifold comes with no preferred inner product. The metric tensor is not something that “appears” automatically; rather, it is a geometric structure we deliberately introduce. The goal of this post is to derive the metric tensor from first principles in a clean, logically economical way.

We begin with a smooth $n$-dimensional manifold $M$. At each point $p\in M$, we want to measure lengths of infinitesimal displacements through $p$. Infinitesimal displacements are modeled by tangent vectors, so the correct object to define is an inner product on each tangent space $T_pM$.

A precise and coordinate-free definition of tangent vectors is the following. A tangent vector at $p$ is a derivation at $p$: a linear map $X:C^\infty(M)\to\mathbb R$ satisfying the Leibniz rule $X(fg)=f(p)X(g)+g(p)X(f)$. The collection of all such derivations forms a real vector space $T_pM$ of dimension $n$.

Now introduce a coordinate chart $(U,\varphi)$ with $\varphi=(x^1,\dots,x^n)$ and $p\in U$. For each coordinate function, define a derivation by

$$ \left.\frac{\partial}{\partial x^i}\right|_p (f) := \frac{\partial (f\circ\varphi^{-1})}{\partial u^i}\Big|_{\varphi(p)}. $$

These derivations form a basis of $T_pM$. Every tangent vector $X\in T_pM$ can therefore be written uniquely as

$$ X = X^i \left.\frac{\partial}{\partial x^i}\right|_p $$

for real coefficients $X^i$. These coefficients depend on the chosen coordinates, but the vector $X$ itself does not. At this point, nothing like length exists yet. To measure length, we must specify an inner product on each tangent space.

A Riemannian metric on $M$ is a rule that assigns to each point $p\in M$ a bilinear map

$$ g_p : T_pM \times T_pM \to \mathbb R $$

such that:

  1. $g_p$ is symmetric: $g_p(X,Y)=g_p(Y,X)$
  2. $g_p$ is positive definite: $g_p(X,X)>0$ for all $X\neq 0$
  3. The assignment $p\mapsto g_p(X_p,Y_p)$ is smooth whenever $X,Y$ are smooth vector fields

This definition is entirely coordinate-free. The metric is simply a smoothly varying inner product on tangent spaces.

Now fix a coordinate chart $(x^1,\dots,x^n)$. Since $g_p$ is bilinear, it is completely determined by its values on basis vectors. Define

$$ g_{ij}(p) := g_p\!\left(\left.\frac{\partial}{\partial x^i}\right|_p,\left.\frac{\partial}{\partial x^j}\right|_p\right). $$

These functions $g_{ij}$ are smooth, symmetric in $i,j$, and vary from point to point. They are the components of the metric tensor in coordinates.

Given two tangent vectors

$$ X = X^i \frac{\partial}{\partial x^i}, \qquad Y = Y^j \frac{\partial}{\partial x^j}, $$

bilinearity immediately gives

$$ g_p(X,Y) = g_{ij}(p)\, X^i Y^j. $$

This is the familiar coordinate expression of the metric. Importantly, this is not a definition but a representation of the underlying geometric object $g$.

From this formula, the squared length of a tangent vector $X$ is

$$ |X|^2 = g_{ij}(p)\,X^i X^j. $$

Thus, the metric tensor generalizes the Euclidean dot product by allowing the coefficients $g_{ij}$ to vary with position and coordinates.

It is instructive to check how these components transform. If we change coordinates from $x^i$ to $\tilde x^a$, then the basis vectors transform via the chain rule:

$$ \frac{\partial}{\partial \tilde x^a} = \frac{\partial x^i}{\partial \tilde x^a}\frac{\partial}{\partial x^i}. $$

Substituting into the definition of the metric components yields

$$ \tilde g_{ab} = g_{ij}\frac{\partial x^i}{\partial \tilde x^a}\frac{\partial x^j}{\partial \tilde x^b}. $$

This transformation law is exactly what characterizes $g_{ij}$ as the components of a $(0,2)$-tensor field. The metric tensor is therefore not merely a matrix-valued function but a genuine tensorial object on the manifold.

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