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International Journal of Creative and Open Research in Engineering and Management

A Peer-Reviewed, Open-Access International Journal Supporting Multidisciplinary Research, Digital Publishing Standards, DOI Registration, and Academic Indexing.
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ISSN: 3108-1754 (Online)
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Volume 02, Issue 9

Published on: September 2026

TENSOR CONTRACTION RULES FOR THE OPERATIONAL EINSTEIN EQUATION: DUMMY INDICES, TRACES, AND CONSERVED MAPS ON LEAD-TIME GEOMETRY

M. Waghmare R. Ingole N. Futane

Sant Gadge Baba Amravati University, Amravati, Maharashtra, India
DOI:https://doi.org/10.55041/ijcope.v2i9.238

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Abstract

Einstein’s field equation is already a statement about contraction. The Einstein tensor is Ricci minus a metric-weighted trace of Ricci. Ricci itself is a contraction of Riemann. The scalar curvature is a second contraction. Bianchi conservation is a further contraction of the differential Bianchi identity. None of those steps is optional bookkeeping. If a dummy index is unpaired, the object is not a tensor of the rank the equation requires. If two free indices are contracted that should have remained free, the field equation collapses to a scalar and can no longer source a metric. This paper writes the contraction rules that turn Riemann into  and that keep , then reads the same rules on an operational lead-time metric whose stress-energy is inventory, congestion, charge, and blocked lots.

 

Keywords: Einstein summation; dummy index; metric contraction; Ricci tensor; Ricci scalar; Einstein tensor; Bianchi identity; operational metric.

How to Cite this Paper

Waghmare, M., Ingole, R. & Futane, N. (2026). Tensor Contraction Rules for the Operational Einstein Equation: Dummy Indices, Traces, and Conserved Maps on Lead-Time Geometry. International Journal of Creative and Open Research in Engineering and Management, <i>02</i>(9), 1-9. https://doi.org/10.55041/ijcope.v2i9.238

Waghmare, M., et al.. "Tensor Contraction Rules for the Operational Einstein Equation: Dummy Indices, Traces, and Conserved Maps on Lead-Time Geometry." International Journal of Creative and Open Research in Engineering and Management, vol. 02, no. 9, 2026, pp. 1-9. doi:https://doi.org/10.55041/ijcope.v2i9.238.

Waghmare, M.,R. Ingole, and N. Futane. "Tensor Contraction Rules for the Operational Einstein Equation: Dummy Indices, Traces, and Conserved Maps on Lead-Time Geometry." International Journal of Creative and Open Research in Engineering and Management 02, no. 9 (2026): 1-9. https://doi.org/https://doi.org/10.55041/ijcope.v2i9.238.

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  • •Published on: Sep 30 2026
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