Mark the true (T) or false (F) for the statements involving variational principles, symmetries, and conservation laws. (

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Mark the true (T) or false (F) for the statements involvingvariational principles, symmetries, and conservation laws.
( ) A material particle, subject to the gravitational influenceof a large mass, goes from one initial event to another finalevent, following the trajectory that makes its own timemaximum.
( ) Consider the trajectory of a particle of mass m, in thespace-time of the Minkowski metric. From the principle of maximumproper time, the dimensionless quantity that is conserved,associated with energy, is the ratio dt/dτ involving the temporalcoordinate and the proper time. However, this ratio is an intensivequantity, whereas energy must be extensive and proportional tomass.
( ) The fact that the coefficients of a given metric have anexplicit dependence on time leads to the law of conservation ofenergy.
( ) The fact that the coefficients of a given metric do not show anexplicit dependence on Cartesian spatial coordinates leads to thelaw of conservation of linear momentum.
( ) Consider a metric written in terms of the spherical coordinatesr, θ and ϕ , where 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π. Assume that none ofthe coefficients of this metric explicitly depend on the ϕcoordinate, although some of them depend on the θ coordinate. Inthis case we can conclude that the angular momentum for rotationsinvolving the ϕ coordinate is conserved.