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By Marco Lanzagorta
One of many significant clinical thrusts lately has been to attempt to harness quantum phenomena to extend dramatically the functionality of a large choice of classical details processing units. specifically, it truly is more often than not approved that quantum pcs and communique platforms promise to revolutionize our details infrastructure.
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Replace standard derivatives with covariant derivatives. 4. Replace the non-invariant volume element d4x with the invariant volume pffiffiffiffiffiffiffi element Àg d4 x. The resulting equation will be generally covariant and will remain valid in the presence of a gravitational field6. For example, the four-velocity of a massive particle is given by: dxμ ð2:133Þ uμ ¼ dτ with the normalization: uμ uμ ¼ À1 ð2:134Þ which is a result of imposing: dS 2 ¼ Àdτ2 ð2:135Þ Then, Newton’s Law in the absence of gravitation can be written as: duμ ð2:136Þ fμ ¼ m dτ where f μ describes non-gravitational forces.
Indeed, it is not possible in general relativity to define a vector as an arrow that stretches from one point towards another point in a curved manifold [1]. A local coordinate basis of the tangent space formalizes the notion of vectors parallel to some determined coordinate axes [1, 6]. V V @α ¼ V μ@μ ¼ V @xν @~ xμ ð2:131Þ where we have used the transformation rules for the contravariant vector Vμ and the covariant local basis element @ μ. In this case, the vector V remains unchanged, but its components Vμ change under a general transformation of coordinates.
Instead, the metric tensor is defined up to a local Lorentz transformation. Indeed, if we transform the tetrad field: eaμ ðxÞ ! e~aμ ð~ x Þ ¼ Λba ðxÞ ebμ ðxÞ ð2:192Þ then the metric tensor remains invariant: gμν ðxÞ ! g~μν ð~ xÞ x Þ e~bν ð~ x Þ ηab ¼ e~aμ ð~ ¼ Λac ecμ ðxÞ Λbd edν ðxÞ ηab ¼ Λac Λbd ηab ecμ ðxÞ edν ðxÞ ¼ ecμ ðxÞ edν ðxÞ ηcd ¼ g μν ðxÞ ð2:193Þ In other words, a tetrad representation of a specific metric is not unique, and different tetrad fields will lead to the same metric tensor if they are related by local Lorentz transformations [23].