Time Dilation on Earth: Why Hemisphere Matters

[A.T.]

We consider points on Earth with a separation. Those are not inertial, and they will feel different effects from the sun.

Points on the Sun's surface are also non inertial. So I still don't see why you cannot say "the sun orbits earth".

 

[A.T.]

Ok. I guess my only point then is that there are better approximations to IF within the solar system.

In GR you cannot approximate an inertial frame for the solar system, unless you want to ignore gravity altogether, but then you cannot explain orbits. You seem to use Newtonian notions of inertial frames.

 

[1977ub]

In GR you cannot approximate an inertial frame for the solar system, unless you want to ignore gravity altogether, but then you cannot explain orbits. You seem to use Newtonian notions of inertial frames.

I think that "approximate" is exactly what you *can* do in the solar system... you just have to keep in mind that is what you are doing. It's true I wasn't taking gravitation into account.

https://en.wikipedia.org/wiki/International_Celestial_Reference_Frame

The International Celestial Reference Frame (ICRF) is a quasi-inertial reference frame centered at the barycenter of the Solar System, defined by the measured positions of 212 extragalactic sources (mainly quasars). Although general relativity implies that there are no true inertial frames around gravitating bodies, the ICRF is important because it definitely does not exhibit any measurable angular motion since the extragalactic sources used to define the ICRF are so far away. The ICRF is now the standard reference frame used to define the positions of the planets (including the Earth) and other astronomical objects. It has been adopted by International Astronomical Union since 1 January 1998. ICRF had a noise floor of approximately 250 microarcseconds (µas) and an axis stability of approximately 20 µas; this was an order-of-magnitude improvement over the previous Fifth Fundamental Catalog (FK5)

 

[PeterDonis]

In GR you cannot approximate an inertial frame for the solar system, unless you want to ignore gravity altogether, but then you cannot explain orbits.

Actually, you can, by using a weak-field approximation in which gravity is viewed as a tensor field on a flat spacetime background. This is basically what the ICRF that 1977ub mentioned does. It is called "quasi-inertial" because it is an inertial frame with respect to the flat spacetime background, not with respect to the full curved spacetime (as you note, the solar system is too large for any local inertial frame with respect to the full curved spacetime to cover it). Gravity is weak enough everywhere in the solar system that this works fine.

A similar frame can be constructed centered on the Earth, called an Earth-Centered Inertial (ECI) frame:

http://en.wikipedia.org/wiki/Earth-centered_inertial

The difference here is that, since the Earth only takes a year to orbit the Sun, while the solar system takes about 250 million years to orbit the center of the galaxy, an ECI frame has non-negligible acceleration (i.e., is not really inertial) on many time scales of interest (the Wikipedia page talks about this). But it's still basically the same thing as above: gravity is weak enough everywhere in Earth's vicinity that it can be treated as a tensor field on a flat spacetime background, and you can construct "quasi-inertial" frames that are inertial relative to the flat spacetime background.

 

[1977ub]

Points on the Sun's surface are also non inertial. So I still don't see why you cannot say "the sun orbits earth".

Heliocentric and geocentric are models, not truths. So you can use either, depending upon your practical purpose. Most people balk at using the geocentric model to describe the motions of solar system bodies.

https://en.wikipedia.org/wiki/Deferent_and_epicycle

 

[mfb]

Points on the Sun's surface are also non inertial.

Yes but the influence of Earth on them is 6 orders of magnitude smaller than the other direction.

 

[harrylin]

Ok. I guess my only point then is that there are better approximations to IF within the solar system.

Hmm. If we are not speaking in absolutes, then every frame such as center of Earth or even a point on Earth's surface is a "local inertial frame" (albeit not a very "good" one)?

Clarifying a little more: PeterDonis isn't talking about Galilean frames but about the "local inertial frames" of GR. Those are free-fall frames that mimic inertial (Galilean) frames locally - for example the inside of a freely falling elevator is such a frame. The "Earth Centered Inertial frame" is (even when correcting for its orbit) not such a "local inertial frame" because objects are accelerated towards the center. It is only far away from heavy mass that "inertial frames" and "local inertial frames" match with each other. And many people who are versed in GR mean such "local inertial frames" when they speak of "inertial frames".

 

[PeterDonis]

The "Earth Centered Inertial frame" is (even when correcting for its orbit) not such a "local inertial frame" because objects are accelerated towards the center.

Correct; the terminology is somewhat confusing but it's the standard terminology so we just have to deal.

 

[stevendaryl]

When I say the speed is zero, I mean nothing more or less than that the distance between them is not changing with time.

What you're talking about is sometimes called the "closing speed" (the rate at which the distance between objects changes). That's not the same thing as "relative velocity", which is the quantity that is important for relativity.

On a piece of graph paper, set up the x-axis horizontally and y-axis vertically. Draw a circle centered on x=0, y=0 of radius 1 inch. Now, imagine two ants on opposite sides of the diameter: one at x=+1, y=0, the other at x=-1, y=0. They start moving around the circle counterclockwise at a rate of 1 inch per second.

Then the distance between the ants never changes. But their relative velocity is not zero. The ant on the left, at x=-1 has initial velocity -1 in the y-direction. The ant on the right, at x=+1 has initial velocity +1 in the y-direction. Their velocities are not the same. The relative velocity has magnitude 2 inches/second.

 

[stevendaryl]

Why? The centers of both are inertial.

Well, there is a quantitative difference between the sun-centered frame and an Earth-centered frame, which is that a sun-centered frame has approximately a constant metric, while an Earth-centered frame has a time-dependent metric (that is, the components of the metric are time-dependent). Both metrics are time-dependent, but in the case of the sun-centered case, the time-dependence is a small correction.