J.C.N. Smith,
The nature of time is an interesting subject. I don't get very partisan over the issue of time existing, for physics seems not to welcome ontological or existential ideas from the outset, but only suggests these within some theoretical construct. We have some issues with reality in a quantum mechanical content, in particular with nonlocality and a local definition of reality. Time and space though are not impacted by this. However, it is likely that quantum gravity will have implications along these lines. Julian Barbour is pretty much into the notion that time does not exist. This is based largely on the Wheeler DeWitt equation HΨ[g] = 0, which is a quantum version of the Hamiltonian constraint in ADM relativity.
I could well enough imagine presenting how time exists, but space does not. We could presume there is some one dimensional space, a line or curve, and there is a fibration on that space by a three dimensional space. This internal space is a symmetry of the dynamics of this one dimensional parameterized space we label as time. This then connects to relativity when we consider the metric line element
ds^2 = -c^2dt^2 g_{ij}dx^idx^j,
where mixed time-space metric components are not included. We have here two notions of time. The first is the proper time τ = s/c, which is the invariant of relativity. The other time is a coordinate time t, which is not an invariant.
The obvious question to ask is whether ds is real. We can multiply it by mc^2 and define an action according to the extremal principle of the proper interval
S = mc∫ds,
which appears real in some sense. It has units of action, or angular momentum, which is a measurable quantity. Yet there is something a bit troublesome about all of this. How does the observer on this world line actually measure this interval? A clock is employed which must have some system of oscillations, such as a spring. Yet this is measuring the invariant interval according to something carried on that world line that deviates from the world line. Hence some sort of nongeodesic motion is being used to define or measure an interval along a geodesic path. Of course I am thinking primarily of a mechanical clock, but an atomic one still appears to hold for an EM field must be applied to knock electrons in the Ce atoms.
This Lagrangian is measured according to something which is not invariant. So we might then consider that action as dS = pdq - Hdt. Now we have some Hamiltonian, which might include a part for the dynamics of the clock. Hamiltonians must be specified on some Cauchy surface of data with a coordinate time direction. Yet this has gotten us into some funny issue, for to define an invariant interval it appears that we need a coordinate defined clock.
So far we have some identification of Hdt, or the square of this, with the c^2dt^2 in the interval above. We then have that the bare action term ∫pdq is identified with
∫pdq = mc sqrt{g_{ij}dx^idx^j}.
So we have a bare action given by our fibration, but we also have some constraint, where H acts as a Lagrange multiplier. So we then have our one dimensional curve defined in a spacetime, where the space is the space of fibration and the Lagrange multiplier determines the symmetry of that fibration which is the Lorentz group.
Now to make things curious, we could imagine this picture as dual in some ways to the picture where time does not exist, but space does. The duality might then have a noncommutative coordinate geometric content in quantum gravity.
The acceleration can be found by a number of means. F = ma with the Newtonian law of gravity and the centripetal acceleration a = v^2/r,
mv^2/r = -GMm/r^2.
This gives v = sqrt{GM/r}, which is v = 29.5km/s or v = 2.95x10^4m/s, for the mass of the sun and r = 1.5x10^8km. The acceleration is then 5.8m/s^2.
As I indicated I think there is some noncommutative geometric issue with the nature of space and time in quantum gravity. This will be a noncommutative geometry that is more general than the Kahler geometry of geometric quantization. That extends the pseudo-complex symplectic structure of classical mechanics into a complex structure with a Hermitian complentarity between conjugate variables. A simplectic group with z_i = (q_i, p_i} (index notation implied) obeys dz_i/dt = Ω_{ij}z_j. In quantum mechanics this is generalized to a commutation system, and the symplectic 2-form implies an operator valued Hamiltonian. However, this is not the most general system possible. This may be extended to in noncommutative geometry with more general Usp(n) groups. A unitary Lie structure can give rise to commutators [q_i, q_j] = ħω_{ij}, which are necessary in string theory with uncertainty principles involving transverse and longitudinal string modes ΔX^ΔX^- ~ L_s = 4πsqrt{α'}, α' = string parameter and L_s the string length. Noncommutative geometry is then a setting for complementarity principles.