Geodesic Equation: Understanding Proper Time & x^α

[Silviu]

Hello! I am a bit confused about the geodesic equation. So for a massive particle it is given by: $\frac{d}{d\tau}\frac{dx^\alpha}{d\tau}+\Gamma^\alpha_{\mu\beta}\frac{dx^\mu}{d\tau}\frac{dx^\beta}{d\tau}=0$, where $\tau$ is the proper time, but in general can be any affine parameter. I am confused about $x^\alpha$. In which coordinate is this measured i.e. where is the observer located. I am actually doing the Schwarzschild metric now, and from the geodesic equation you can get $\frac{dx^\alpha}{d\tau}$, but I am not sure for who do you get this, as all the values of the geodesic equations are evaluated at the position of the moving object, so for example $\frac{dt}{d\tau}$, connects the time of the observer, with the proper time of the object, however, different observers, in Schwarzschild metric, measure different times, yet the geodesic equation gives just a value for it. Can someone explain this to me please? Thank you!

 

[pervect]

You can use any coordinates you wish, that's the point. For the Schwarzschild metric, you'd use the Schwarzschild coordinates. At any particular point, you use the values of $\Gamma^{\alpha}{}_{\mu}{\gamma}$ and $\frac{dx^i}{d\tau}$ at the point you select to write down the equation. This isn't associated with any particular observer - it doesn't need to be, it only needs to be associated with a coordinate system. In other words, if you know r, theta, phi, and t as a function of proper time $\tau$, you know the trajectory of the object in the Schwarzschild coordinates. The only thing associated with the object is the proper time $\tau$. You are not limited to selecting any particular object, it works for all objects.

I'm not even sure we mean the same thing by "an observer", I'm guessing you might be reaching towards using a set of orthonormal basis vectors at some point as defining an "observer" at that point. This is a very useful and powerful technique (if that's what you're doing, or trying to do) - but it doesn't have anything to do directly with the geodesic equation which doesn't need an observer, just coordinates. Or perhaps you are asking about which observer to use the proper time of - again, you can use any observer, or any affine parameter.

 

[Silviu]

You can use any coordinates you wish, that's the point. For the Schwarzschild metric, you'd use the Schwarzschild coordinates. At any particular point, you use the values of $\Gamma^{\alpha}{}_{\mu}{\gamma}$ and $\frac{dx^i}{d\tau}$ at the point you select to write down the equation. This isn't associated with any particular observer - it doesn't need to be, it only needs to be associated with a coordinate system. In other words, if you know r, theta, phi, and t as a function of proper time $\tau$, you know the trajectory of the object in the Schwarzschild coordinates. The only thing associated with the object is the proper time $\tau$. You are not limited to selecting any particular object, it works for all objects.

I'm not even sure we mean the same thing by "an observer", I'm guessing you might be reaching towards using a set of orthonormal basis vectors at some point as defining an "observer" at that point. This is a very useful and powerful technique (if that's what you're doing, or trying to do) - but it doesn't have anything to do directly with the geodesic equation which doesn't need an observer, just coordinates. Or perhaps you are asking about which observer to use the proper time of - again, you can use any observer, or any affine parameter.

I am still confused. Let's say we have an object moving on a geodesic on a circle, around a star in the Schwarzschild metric and I want to calculate the period. Now an observer at infinity and another one close to the black hole, would measure different periods for the moving body, in their respective reference frame (time passes slower closer to the star). So if at infinity the period would be 1 year, while the observer closer to the star would measure (let's say) half a year for the same object moving around the star. So the t of the observer at infinity, would be different than the t of the observer closer to the star. Yet the geodesic equation has just one t ($x^0$). This is what confuses me, whose t is that?

 

[Nugatory]

So the t of the observer at infinity, would be different than the t of the observer closer to the star.

If they're both using Schwarzschild coordinates then they're both using the same $t$ coordinate. However, only the observer at infinity will find that those $t$ values match up to the time ticked off by his wristwatch (which is to say the coordinate-independent proper time between successive ticks). That's inconvenient for observers nearer the star, but it doesn't change the basic nature of the solution to the geodesic equation: A geodesic is a particular set of events in spacetime, namely the ones that a free-falling test mass moves through. These events will have different labels (the four $x^\mu$) depending on the coordinate system you choose, but whatever coordinate system you choose, the geodesic equation will properly calculate the coordinates in that coordinate system of the events along the geodesic.

 

[pervect]

I am still confused. Let's say we have an object moving on a geodesic on a circle, around a star in the Schwarzschild metric and I want to calculate the period. Now an observer at infinity and another one close to the black hole, would measure different periods for the moving body, in their respective reference frame (time passes slower closer to the star). So if at infinity the period would be 1 year, while the observer closer to the star would measure (let's say) half a year for the same object moving around the star. So the t of the observer at infinity, would be different than the t of the observer closer to the star. Yet the geodesic equation has just one t ($x^0$). This is what confuses me, whose t is that?

The geodesic equation has $\tau$, which is a proper time, and t, which is a coordinate time. When you talk about "t of the observer", I'm not sure which of the two you are talking about, or if you are talking about something else. I would tend to assume that you are talking about the Schwarzschild coordinate t, though, rather than the proper time $\tau$ (tau).

If you put a clock in the circular orbit, you can calculate the period as read by that orbiting clock when it returns to the same angular coordinate it started with. This is a coordinate independent quantity, and doesn't depend on any clock synchronization conventions. As you hopefully recall, in special relativity, different observers have different notions of simultaneity. Perhaps you do not recall this, if you do not recall this it is worth studying and/or reviewing, because it may shed light on your confusion. It would also be good to review the difference between proper time and coordinate time.

As I remarked, the proper time, which can be regarded as the time read out by a wristwatch carried by someone orbiting the black hole, is independent of the choice of coordinate or notions of simultaneity. It's independent of simultaneity because the same clock is present at the "start or orbit" and "end of orbit" events. There is another sort of "period" you can calculate which does depend on your choice of coordinates and/or your simulataneity conventions.

You can calculate a different time, by finding the difference in Schwarzschild t coordinates. There is an event, which I'll call "start of orbit", and another event, which I'll call "end of orbit". If you find the t-coordinate in Schwarzschild coordinates of both events, and take the difference, you get a number which is the time it takes for the orbit in Schwarzschild coordinates.

My best advice is to forget about the notion of an observer in General Relativity, as the concept is rather vague. But perhaps you will insist. If you insist, you need to figure out what YOU mean by "an observer". The general answer the physics gives you is that the notion is ambiguous, because the notion of simultaneity is ambiguos, as I remarked earlier.

If we imagine we have a clock following some arbitrary worldline, and we have a notion of simultaneity, I would suggest that the notion of "the time according to that observer" would be the time reading on that clock simultaneous with the event ""end of orbit" minus the time reading on that clock simultaneous with the event "start of orbit". But it's up to you to define a worldine, AND that worldine's notion of simultaneity.

Most notions of simultaneity you might choose be translated into a coordinate system, where simultaenous events are given the same time coordinate. (There are some possible issues with this statement that I would rather avoid at the current time, but if necessary we can get into that). If we assume that your notion of simultaneity can be translated by defining an appropriate set of coordinates, the challenge then is to find the coordinate system you want to use. that define "an observer". There are some choices in the literature that you could investigate, such as Fermi normal coordinates, but it's a rather advanced topic. The usual recommendation is to realize that calculating coordinate dependent quantities isn't really necessary, because the choice of coordinates is a human convention and can't affect the physics, just how one describes it. Fermi normal coordinates may provide some insight as to a scheme for defining simultaneity, though. It would be long and technical to talk about Fermi normal coordinates, but if you're interested, you can ask specifically and I'll try.

 

[Yukterez]

dxⁱ/dτ=ẋⁱ is not the local velocity (c=1), but, in the case of Schwarzschild coordinates, rather the lengths as measured by the observer at infinity differentiated by the proper time of the local free falling observer (c=∞). The general relation is

$\rm \dot x^i = \frac{v^i }{ \sqrt{(1 - v^2) \ |g_{i i}|}} - \dot t \ \frac{g_{t i}}{g_{i i}}$

where vⁱ is the local velocity component in the i-direction and v the total local velocity.

 

[Mordred]

The way I learned to understand geodesic equations is by looking at the specifics of "coordinate basis" via the Kronecker and Levi-Cevita. The choice of coordinates is arbitrary which is the entire point. So one must look at the affine connections.