Static artifact

Metric → Tensor Calculator

A planned metric-first workflow for deriving geometric tensors from a specified spacetime line element.

G_{\mu\nu}

What the tool does

The calculator is intended to accept a metric ansatz, organize the components, and expose intermediate steps toward Christoffel symbols, curvature tensors, and Einstein tensor terms.

Why it matters

For general relativity work, a reproducible tensor pipeline reduces transcription errors and makes it easier to audit each symbolic step before interpreting a physical result.

Connection to CTC paper

Yuvan's CTC paper depends on tracing how metric choices affect causal structure and energy-condition checks. This tool frames those calculations around the metric components and the tensor quantities they generate.

Research role

This page will document the symbolic path from a metric tensor to curvature-derived quantities, with emphasis on assumptions, coordinate conventions, and independently checkable intermediate outputs.

Near-term build

The first implementation should focus on a small set of validated metrics before accepting arbitrary user input. The priority is correctness and traceability, not breadth.

Static Validation Artifact

Gödel Metric Pipeline

This page documents the computational pipeline and the validation checks required before publishing computed tensor outputs. The example is the Gödel metric workflow used around Yuvan's closed-timelike-curve paper, but this artifact does not publish full Christoffel, curvature, or Einstein tensor components until a notebook or script verifies them component by component.

Example input diagnostic

g_{\phi\phi}^{\mathrm{G\ddot{o}del}} = \sinh^2(r) - \sinh^4(r)

Step 1

Metric input

g_{\mu\nu}(x)

Record the coordinate chart, sign convention, parameter normalization, and source for the Gödel metric before any tensor manipulation. The currently displayed verified scalar diagnostic is the angular component used by the CTC visualizer.

Step 2

Inverse metric

g^{\mu\nu}

Compute the symbolic inverse only after the metric matrix and conventions are fixed. The inverse is not published here until the consistency identity is checked in the same convention.

Step 3

Christoffel symbols

\Gamma^{\rho}_{\mu\nu} = \tfrac{1}{2}g^{\rho\sigma}(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu})

Use the verified metric and inverse to derive connection coefficients. This stage should publish intermediate assumptions and one independently checked component before claiming broader tensor output.

Step 4

Curvature / Einstein diagnostic

G_{\mu\nu}

Curvature and Einstein tensor diagnostics should be treated as validation targets, not decorative output. They require source comparison, convention notes, and explicit component-level checks.

Validation checklist

Metric signature check

Confirm the intended spacetime signature and state it beside the metric definition.

Metric symmetry check

g_{\mu\nu} = g_{\nu\mu}

Verify symmetry before attempting inversion or connection coefficients.

Inverse consistency

g_{\mu\nu}g^{\nu\rho} = \delta^{\rho}_{\mu}

Check that the symbolic inverse contracts back to the identity tensor in the chosen coordinate convention.

Known Gödel threshold cross-check

g_{\phi\phi} < 0 \quad \mathrm{for}\quad r > \ln(1+\sqrt{2})

Use the known chronology threshold as a scalar sanity check before publishing higher-rank tensor quantities.

Units and convention notes

Document normalization choices, coordinate ordering, index order, and any sign convention for curvature tensors.

Next reproducibility step

Publish a notebook or script showing one explicit verified tensor component, then link it from GitHub and the Validation Archive once the calculation is reproducible.

Research caution

Static validation artifact only. It documents the computational pipeline and checks required before publishing computed tensor outputs; it is not yet a live symbolic engine.