Understanding the Role of Tensor Calculus in General Relativity

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SUMMARY

Tensor calculus is essential for understanding General Relativity (GR), as it provides the mathematical framework necessary for expressing the theory. Without tensor calculus, one cannot fully grasp the complexities of GR, which is fundamentally rooted in differential geometry. Key mathematical prerequisites include multivariable real calculus, linear algebra, point set topology, and the ability to solve ordinary differential equations (ODEs). The discussion emphasizes that tensor calculus is not merely an accessory but a foundational element of GR.

PREREQUISITES
  • Multivariable real calculus
  • Linear algebra
  • Point set topology
  • Solving ordinary differential equations (ODEs)
NEXT STEPS
  • Study the fundamentals of tensor calculus
  • Explore differential geometry concepts
  • Learn about the Einstein field equations in GR
  • Investigate the role of non-linear partial differential equations in physics
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Students and researchers in physics, mathematicians focusing on geometry, and anyone interested in the mathematical foundations of General Relativity.

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In an attempt to solve the mystery of dark energy, I came across problems concerned with the General Relativity. In it, I observed that many of the problems were related with the tensor calculus.
I want to know that what importance does tensor calculus hold in GR? Are there any other fields of mathematical that too play an important role in the subject?
 
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Your intentions are bold, for sure. You cannot conceive General Relativity in the absence of tensor calculus. Usually tensor calculus is seen as a part of differential geometry, so any mathematical prerequisites of diffgeo are mandatory: multivariable real calculus, linear algebra and a little point set topology + solving ODEs (to which the non-linear PDEs of GR can be usually reduced).
 
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General relativity is expressed in tensor calculus. You can't have GR without it.

I guess the analogy would be "what importance does English have to Shakespeare's plays?"
 

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In an inertial frame of reference (IFR), there are two fixed points, A and B, which share an entangled state $$ \frac{1}{\sqrt{2}}(|0>_A|1>_B+|1>_A|0>_B) $$ At point A, a measurement is made. The state then collapses to $$ |a>_A|b>_B, \{a,b\}=\{0,1\} $$ We assume that A has the state ##|a>_A## and B has ##|b>_B## simultaneously, i.e., when their synchronized clocks both read time T However, in other inertial frames, due to the relativity of simultaneity, the moment when B has ##|b>_B##...
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