# I Time in Space-time

1. Sep 14, 2016

### andrew s 1905

I am interested in the current understanding of time in SR GR QM and Cosmology and so apologies if this is the wrong place to post.

I am currently reading "The End of Time" by Julian Barbour and am aware of "From Eternity to Here" by S Carroll and " The Singular Universe and the Reality of Time" by R M Unger and L Smolin. I have also read the various PhysicsForums Insights e.g. on the Block Universe

I realise the books are semi popular expositions of possibly non-standard views. In view of this I would be please to be pointed at any modern discussions or treatments especially if they include observable consequences. I am not particularly interested in interpretational issues as say with the various interpretations of QM.

Thanks Andrew

2. Sep 14, 2016

### Orodruin

Staff Emeritus

3. Sep 14, 2016

### andrew s 1905

Sorry the question is are there any modern discussion (papers, books etc.) that cover the role of time plays in physics and if so can you provide the references?

I don't have one specific question, hence the request for references, but by way of an example of the issues I am interested in, is the treatment of time as a parameter (say in QM or SR) rather than as a coordinate equivalent to the space dimensions physically meaningful and if so would this have observable effects.

If this is not a legitimate request or two general then please close the thread.

Regards Andrew

4. Sep 14, 2016

### Staff: Mentor

That was one of the single most disappointing books that I ever read.

Well, there are some distinct concepts of time. One would be "coordinate time", which as you might guess is a coordinate. Another would be "proper time", which is often used to parameterize a worldline. Both concepts of time are used in relativity, so I don't think that we can give you a "rather than" reference. Instead, both are useful in their own right.

5. Sep 15, 2016

### andrew s 1905

Being retired I guess I have time to waste! Any particular reason you felt it so worthless?

I realise there are different concepts of time that's what I am interested in understanding (how may there are why we need them and their interrelation etc. given more or less a clock is a clock) but thanks for taking the time to reply Dale I do appreciate it.

Regards Andrew

6. Sep 15, 2016

### Staff: Mentor

I kept on expecting him to describe how physics could be done without time, and all he did was rename the time variable.

As far as I know there are just those two, coordinate time and proper time. Proper time is the invariant spacetime interval along a given worldline, which is also the time measured by an ideal clock on that worldline. Coordinate time is whatever the coordinate system defines it to be. Often the coordinates are designed so that the coordinate time matches thw proper time for some particular group of worldlines.

7. Sep 15, 2016

### A.T.

Usually some with constant spatial coordinates.

8. Sep 15, 2016

### andrew s 1905

Thanks all for your inputs. I am content I understand the difference between co-ordinate and proper time. As to the other issues that exercise me I suspect they are more on the ontology of time so not for this forum.

Regards Andrew

9. Sep 15, 2016

### vanhees71

There's no principle difference between coordinate and proper time. Given a timelike congruence you define a coordinate time. A time like congruence can be interpreted as a flow of particles, and then this time is the proper time of these particles along these trajectories.

10. Sep 16, 2016

### pervect

Staff Emeritus
I may be being over-literal, but I don't see how a time like congruence alone defines a unique coordinate time. For instance, with a one-parameter group of geodesic time-like congruences, P(s,t), one can re-parameterize the same congruence in different ways with respect to t, i.e, one can implicitly define a new time coordinate t' by the relationship t = f(t',s), then write P(s,t) = P(s,f(t',s)), thus defining P(s,t') via the pullback over f (as long as f has the appropriate inverse).

The special thing about geodesic congruences (IIRC - I should probably look it up to double-check, but I didn't) is that if $\partial_s$ is orthogonal to $\partial_t$ at (s,t), it will remain orthogonal for (s,t+$\delta$) for all $\delta$. I don't believe this holds in general though, so it won't serve as a general prescription for how to parameterize a particular congruence.

11. Sep 17, 2016

### vanhees71

True. Of course you can redefine the so defined coordinate times with arbitrary 1D diffeomorphisms. I.e., if $t$ is one such time parameter, you can as well choose $t'=f(t)$ with $f$ a strictly monotonous differentiable function (usually you choose it monotonously increasing to keep the orientation of the times the same).

12. Sep 17, 2016

### Staff: Mentor

But then the proper time and the coordinate time are different.

13. Sep 17, 2016

### Staff: Mentor

Yes, this is true, because parallel transport preserves the norms and inner products of vectors. But it is independent of how you choose the affine parameter along the geodesic; choosing a new affine parameter changes the numerical magnitude of $\partial_t$, which rescales all nonzero inner products, but it doesn't change which vectors it's orthogonal to (because rescaling an inner product of zero doesn't change it, it's still zero).

I'm not sure exactly what conditions are preserved by Fermi-Walker transport as opposed to parallel transport. In some special cases, at least, F-W transport preserves orthogonality of particular pairs of vectors; for example, it preserves the orthogonality of the basis vectors $\partial_t$ and $\partial_x$ in Rindler coordinates along the worldline of an object with constant proper acceleration.

14. Sep 19, 2016

### vanhees71

In special relativity the Fermi-Walker transport is most easily defined as a description of the transport of arbitrary vectors along a time like curve such that the tetrad transported by an observer on this curve (it's an arbitrarily accelerated observer) such that his spatial basis vectors are not rotating.

This description in words is not very precise, but it must be made clear by the mathematical formulation. Without loss of generality we'll parametrize the world line by it's proper time, $\tau$, such that the four-velocity of the observer is normalized
$$u^{\mu} = \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=\mathrm{d}_{\tau} x^{\mu}, \quad u_{\mu} u^{\mu}=1.$$
Then the momentary proper acceleration of the observer
$$a^{\mu}=\mathrm{d}_{\tau} u^{\mu}$$
is perpendicular to $u^{\mu}$ since
$$u_{\mu} u^{\mu}=1 \; \Rightarrow \; u_{\mu} \mathrm{d}_{\tau} u^{\mu}=u_{\mu} a^{\mu}=0.$$
So it defines a space-like direction.

Now you can define instantaneous inertial restframes of the observer by introducing a special tetrad, i.e., four orthonormal basis vectors as a function of $\tau$, where always $\boldsymbol{e}_0=\boldsymbol{u}$. Now there are a lot of possibilities to choose at any proper time three spacelike Minkowski-orthonormal unit vectors orthnormal to $\boldsymbol{u}$. What we, however, usually want is a tetrad where the observer has his spatial basis vectors not rotated against each other. To achieve this we take an arbitrary Minkowski-orthonormal frame and think the corresponding tetrad to be transported along the worldline such that
$$\boldsymbol{e}_{\mu} \cdot \boldsymbol{e}_{\nu}=\eta_{\mu \nu}, \quad \boldsymbol{e}_0=\boldsymbol{u}.$$
Now we have to establish the "non-rotating property" of the spatial tetrad vectors. To do this we define the Fermi-Walker transport. It's heuristically easily found in the following way. Take an arbitrary vector $\boldsymbol{V}$ along the trajectory and define $\boldsymbol{V}(\tau)$ as the transport of $\boldsymbol{V}(\tau=0)$ along the curve such that when going from $\tau$ an infinitesimal step further to $\tau+\mathrm{d} \tau$ such that it undergoes only an infinitesimal rotation free Lorentz boost in the $\boldsymbol{u}$-$\boldsymbol{a}$ plane. This means that
$$\mathrm{d} V^{\mu}=\mathrm{d} \tau (a^{\mu} u^{\nu}-u^{\mu} a^{\nu}) V_{\nu}$$
or the Fermi-Walker transport is defined by the differential equation
$$\mathrm{d}_{\tau} V^{\mu}=(a^{\mu} u^{\nu}-u^{\mu} a^{\nu}) V_{\nu}.$$
Since the infinitesimal change corresponds just to a Lorentz trafo in the $\boldsymbol{u}$-$\boldsymbol{a}$ plane, which is a plane spanned by a time-like and a space-like vector, and thus the Lorentz trafo is a rotation-free boost. It also has the other properties we like. Indeed, the Fermi-Walker transport of $\boldsymbol{u}$ gives
$$\mathrm{d}_{\tau} u^{\mu} = a^{\mu},$$
because $u_{\mu} u^{\mu}=1$ and $u_{\mu} a^{\mu}=0$.

Further if $V$ and $W$ are Fermi-Walker transported we have
$$\mathrm{d}_{\tau} (\boldsymbol{V} \cdot \boldsymbol{W})=(\mathrm{d}_{\tau} \boldsymbol{V}) \cdot \boldsymbol{W} + \boldsymbol{V} \cdot \mathrm{d}_{\tau} \boldsymbol{W}\\=\boldsymbol{a} \cdot \boldsymbol{W} \boldsymbol{u} \cdot \boldsymbol{V} - \boldsymbol{u} \cdot \boldsymbol{W} \boldsymbol{a} \cdot \boldsymbol{V} + \boldsymbol{a} \cdot \boldsymbol{V} \boldsymbol{u} \cdot \boldsymbol{W} - \boldsymbol{u} \cdot \boldsymbol{V} \boldsymbol{a} \cdot \boldsymbol{W}=0,$$
i.e., the Fermi-Walker transport also leaves the Minkowski product between arbitrary vectors unchanged.

So if you start with an arbitrary tetrad at $\tau=0$ and Fermi-Walker transport it along the worldline it stays a tetrad and from one infinitesimal time step to another you have only a rotation free Lorentz boost.

An important physical application is the Thomas precession. Take a particle with spin (historically it was about an electron moving around an atomic nucleus) which is moving along its world line accelerated by an arbitrary force which doesn't apply a torque to the spin. The corresponding equation of motion for the spin is just that of the Fermi-Walker transport of this spin. It turns out that the spin nevertheless precesses, i.e., it rotates, and that's due to the fact that the composition of two Lorentz boosts (if not in the same spatial direction) leads to a Lorentz trafo that's not rotation free, i.e., it consists of a Lorentz boost followed by a rotation. For an electron moving with constant angular velocity along a circle the Thomas-precession frequency is $\omega_{\text{Thomas}}=(\gamma-1)\omega$.

It is also very easy to extend this now to the curved pseudo-Riemannian spacetime of GR. Here, everywhere, where I wrote $\mathrm{d}_{\tau}$ one has to write a covariant derivative $\mathrm{D}_{\tau}$, defined by
$$\mathrm{D}_{\tau} V^{\mu} = \mathrm{d}_{\tau} V^{\mu} + {\Gamma^{\mu}}_{\nu \rho} V^{\nu} u^{\rho}.$$
Particularly the proper acceleration is defined by
$$a^{\mu} = \mathrm{D}_{\tau} u^{\mu}.$$
With this you have a Fermi-Walker transport in curved space time with basically the same geometrical "local" meaning as in flat Minkowski space. It also is clear that Fermi-Walker transport along a time-like geodesic is identical with the parallel transport along this geodesic, because in this case $a^{\mu}=0$. The Fermi-Walker transport can be used to derive the geodesic precession of a spinning top in non-rotating spacetimes (like the Schwarzschild metric) as well as the Lense-Thirring effect on a spinning top in rotating spacetimes (like the Kerr metric).