If the metric itself varies, it could be either because the metric really does vary or . Spacelike geodesics in special relativity are stationary by the above definition. studied. Suppose an observer uses coordinates such that all objects are described as lengthening over time, and the change of scale accumulated over one day is a factor of \(k > 1\). Smooth deformations of a Minkowski type metric in a four-dimensional space-time manifold are considered. For the spacelike case, we would want to define the proper metric length \(σ\) of a curve as \(\sigma = \int \sqrt{-g{ij} dx^i dx^j}\), the minus sign being necessary because we are using a metric with signature \(+---\), and we want the result to be real. Comment: 15 pages, no figure. A world-line is a timelike curve in spacetime. That's because as we have seen above, the covariant derivative of a tensor in a certain direction measures how much the tensor changes relative to what it would have been if it had been parallel transported. Introduction.
For this reason, we will assume for the remainder of this section that the parametrization of the curve has this property. It would therefore be convenient if \(T^i\) happened to be always the same length.
In this case depending on the sign of coupling constant we have either late time accelerated mode of expansion or oscillatory mode of evolution. For example, any spacelike curve can be approximated to an arbitrary degree of precision by a chain of lightlike geodesic segments. The following contents were once discussed in. In our previous article Local Flatness or Local Inertial Frames and SpaceTime curvature, we have come to the conclusion that in a curved spacetime, it was impossible to find a frame for which all of the second derivatives of the metric tensor could be null. The answer is a line. In two other cases the space-time evolves into either LRSBI or FRW Universe. Example \(\PageIndex{2}\): Christoffel symbols on the globe, quantitatively. Given a certain parametrized curve \(γ(t)\), let us fix some vector \(h(t)\) at each point on the curve that is tangent to the earth’s surface, and let \(h\) be a continuous function of \(t\) that vanishes at the end-points. Have questions or comments? This problem seems to be overlooked for a long time. which require fine-tuning. In the spacetime with orthogonal coordinates, the energy momentum tensor T µν is relatively simple. After marching down to the equator, march 90 degrees around the equator, and then march back up to the north pole, always keeping the javelin pointing horizontally and "in as same a direction as possible" along the meridian. However, one can have nongeodesic curves of zero length, such as a lightlike helical curve about the \(t\)-axis.
The most general form for the Christoffel symbol would be, \[\Gamma ^b\: _{ac} = \frac{1}{2}g^{db}(L\partial _c g_{ab} + M\partial _a g_{cb} + N\partial _b g_{ca})\]. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The role of spinor field in the evolution of the Universe is We find in the space-time with intrinsically nondiagonal metric, the orbit of a spinor deviates from the geodesic slightly, so the principle of equivalence is broken by the spinors moving at high speed.
In case of a Bianchi type-III model the space-time remains locally rotationally symmetric all the time, though the isotropy of space-time can be attained for a large proportionality constant. On this foundation, we derive the complete classical mechanics from the dynamical equation and get some interesting results. This topic doesn’t logically belong in this chapter, but I’ve placed it here because it can’t be discussed clearly without already having covered tensors of rank higher than one. In the case where the whole curve lies within a plane of simultaneity for some observer, \(σ\) is the curve’s Euclidean length as measured by that observer. In this report we have considered a polynomial nonlinearity which is a function of invariants constructed from the bilinear spinor forms. In this case for an expanding Universe we have asymptotical isotropization. The situation becomes even worse for lightlike geodesics. effective pressure, which can be seen as an alternative source for late time In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection.
We separate the spinor connection into two parts. In Example \(\PageIndex{1}\), we inferred the following properties for the Christoffel symbol \(Γ^θ\: _{φφ}\) on a sphere of radius \(R: Γ^θ\: _{φφ}\) is independent of \(φ\) and \(R\), \(Γ^θ\: _{φφ} < 0\) in the northern hemisphere (colatitude \(θ\) less than \(π/2\)), \(Γ^θ\: _{φφ} = 0\) on the equator, and \(Γ^θ\: _{φφ} > 0\) in the southern hemisphere. The condition \(L = M\) arises on physical, not mathematical grounds; it reflects the fact that experiments have not shown evidence for an effect called torsion, in which vectors would rotate in a certain way when transported. This is essentially a mathematical way of expressing the notion that we have previously expressed more informally in terms of “staying on course” or moving “inertially.” (For reasons discussed in more detail below, this definition is preferable to defining a geodesic as a curve of extremal or stationary metric length.). there are flows but no sources or sinks of energy-momentum) could be epxressed by saying: Stationarity is defined as follows.
If the geodesic were not uniquely determined, then particles would have no way of deciding how to move. This is described by the derivative \(∂_t g_{xx} < 1\), which affects the \(M\) term. The solution to this chicken-and-egg conundrum is to write down the differential equations and try to find a solution, without trying to specify either the affine parameter or the geodesic in advance. Because we construct the displacement as the product \(h\), its derivative is also guaranteed to shrink in proportion to for small . A curve can be specified by giving functions \(x^i(λ)\) for its coordinates, where \(λ\) is a real parameter.
A negative coupling constant leads to an oscillatory mode of expansion, whereas a positive coupling constant generates expanding Universe with late time acceleration. For a spinor in skew spacetime, the principle of equivalence seems to be broken. The ground state of the dark spinor provides a small negative pressure which may be important in cosmology. If so, then 3 would not happen either, and we could reexpress the definition of a geodesic by saying that the covariant derivative of \(T^i\) was zero. Within the scope of Bianchi type VI,VI0,V, III, I, LRSBI and FRW cosmological models we have studied the role of nonlinear spinor field on the evolution of the Universe and the spinor field itself. What about quantities that are not second-rank covariant tensors? Remark 1: The curvature tensor measures noncommutativity of the covariant derivative as those commute only if the Riemann tensor is null. g_{?? It is also shown that the introduction of the Cosmological constant ($\Lambda$-term) in the Lagrangian generates oscillations of the BI model, which is not the case in absence of $\Lambda$ term. Here the spinors are quantized and identified by nonlinear potentials, and their state functions are derived from standard statistical and variation principles. \(Γ^θ\:_{φφ}\) is computed in example below. Say you start at the north pole holding a javelin that points horizontally in some direction, and you carry the javelin to the equator, always keeping the javelin pointing "in as same a direction as possible", subject to the constraint that it point horizontally, i.e., tangent to the earth. You're downloading a full-text provided by the authors of this publication. We consider the cases when $F$ is the power or trigonometric functions of its arguments. This is the wrong answer: \(V\) isn’t really varying, it just appears to vary because \(G\) does. because the metric varies. The logarithmic derivative of \(e^{cx}\) is \(c\). In the mean field approximation an equation is derived for the critical surface for the coupling constants of the effective fermion action. One thing that the two paths have in common is that they are both stationary. The quasilocal interaction vertices responsible for the formation of dynamic fermion mass are classified for these models in the near-critical region of coupling constants. Why not just define a geodesic as a curve connecting two points that maximizes or minimizes its own metric length? Consistency with the one dimensional expression requires \(L + M + N = 1\).
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