Calculator Input
Formula Used
Coordinate transformation rule:
g′ab = ΣiΣj (∂xi/∂x′a) gij (∂xj/∂x′b)
Matrix form:
g′ = JT g J
Line element:
ds² = g′ab dx′a dx′b
Determinant check:
det(g′) = det(g) × det(J)²
Example Data Table
| Example | Original metric | Jacobian idea | Expected use |
|---|---|---|---|
| Minkowski metric | diag(-1, 1, 1, 1) | Lorentz boost | Check invariant spacetime interval. |
| Polar plane metric | diag(1, r²) | Coordinate basis change | Study radial and angular distances. |
| Spherical slice | diag(1, r², r²sin²θ) | Curvilinear transformation | Review curved coordinate components. |
How to Use This Calculator
- Select the tensor dimension, usually four for spacetime.
- Enter the original metric matrix row by row.
- Enter the Jacobian matrix for old coordinates versus new coordinates.
- Add a displacement vector if you want an interval value.
- Press the calculate button.
- Review the transformed metric, determinant check, inverse matrix, and graph.
- Use CSV or PDF export for reports and records.
Spacetime Metric Transformation Guide
Why Coordinate Changes Matter
A spacetime metric describes how intervals are measured. It links coordinate changes to physical distance, time, and causal type. The same geometry may look different in another coordinate system. This calculator helps you test that change with direct tensor algebra. It is useful for relativity examples, geometry lessons, and symbolic checks before deeper analysis.
Metric Inputs
The original metric should be entered as a square matrix. A flat spacetime example often uses the diagonal form minus one, one, one, one. Curved or curvilinear systems may contain variable dependent values. This tool accepts decimal values, scientific notation, simple fractions, pi, and e. Each row must contain the same number of values.
Jacobian Meaning
The Jacobian stores derivatives of old coordinates with respect to new coordinates. This direction is important. If the opposite Jacobian is entered, the result will describe a different mapping. The determinant check helps reveal many entry mistakes. When the determinant is zero, the transformation is not locally invertible.
Reading the Results
The transformed matrix shows each new metric component. Diagonal values describe squared scale behavior along coordinate directions. Off diagonal values show mixed terms. The interval value classifies the entered displacement as timelike, spacelike, or null under the chosen sign convention. The graph displays component magnitudes for quick comparison.
Practical Checks
Use a known example first. Then replace the metric or Jacobian with your case. Compare determinant values, symmetry error, and interval behavior. Small rounding differences are normal. Large asymmetry usually means a typing error, mismatched dimension, or wrong transformation direction.
FAQs
1. What does this calculator transform?
It transforms a metric tensor from one coordinate basis to another using a supplied Jacobian matrix.
2. Which Jacobian should I enter?
Enter ∂x/∂x′, meaning old coordinate derivatives with respect to the new coordinates.
3. Can I use a 4D spacetime metric?
Yes. Select dimension four, then enter a 4 by 4 metric and matching 4 by 4 Jacobian.
4. What does ds squared mean?
It is the interval from the transformed metric and your entered displacement vector in the new coordinates.
5. Why is the determinant check useful?
It confirms whether det(g′) matches det(g) times det(J) squared, allowing for rounding error.
6. Does the calculator handle symbolic variables?
No. It handles numeric values, fractions, scientific notation, pi, and e for stable calculation.
7. What does symmetry error show?
It reports the largest difference between mirrored metric components, such as g12 and g21.
8. Can I export the answer?
Yes. Use the CSV button for spreadsheet data or the PDF button for a clean report.