Speady said:
Slower in both directions?
If the traveler looks at my clock with a telescope, then, looking through his telescope, he can just as well count how many circles the Earth has revolved around the sun. After all, the movements of the clock and of the orbiting Earth are in a fixed relationship to each other, aren't they? No matter how far and how fast the traveler goes: it cannot be the case that he counted more or fewer laps than the person who stayed at home? Surely it cannot be that one says "3 rounds" and the other says "5 rounds"? During the experiment, the Earth could only have made one and the same number of circles.
Okay: I can understand that the traveler counts fewer rounds on the way there, because of the delay, because of the time it took for the light from my surroundings to reach him. But on the way back it seems logical to me that he sees the orbiting Earth as an accelerated image, due to the ever shorter time it takes for the light from my surroundings to reach him. So at the end the number of laps is the same again. We call turning one circle "the passing of a year". So, isn't it the case that for the traveler and me the same number of years have passed?
I hope I don't get blocked right away after asking this question.
Usually you get this confusion about space-time measurements solved when remembering that what's simultaneous in one inertial reference frame is not simultaneous in another.
This already starts with the clock synchronization, which by definition is made only between clocks all at rest wrt. one inertial reference frame. By definition you think about one standard clock at the origin of this frame and any other standard clock at another arbitrary point at rest relative to the clock at the origin at a distance ##r## from the origin. Then you can synchronize both clocks by sending a light signal to the other clock being reflected there. Now you can measure the time needed from sending the signal to receiving the reflected one. This is a measurement at one position, i.e., the origin of the frame and thus really feasible in the lab. By definition of the time units (like the second in the SI units) this time is ##2 \Delta r/c##, where ##c## is the (arbitrarily defined as in the SI!) speed of light. Now by definition it's assumed that the signal takes as long to go from the origin to the clock to be synchronized as the reflected signal needs to be reflected back (again note that this is assumed in an inertial reference frame and for clocks being both at rest in this frame!). Thus to synchronize the distant clock with the clock at the origin. You have to preset the distant clock to a time ##r/c## and send the signal from the origin at ##t=0##, and start the distant clock as soon as the light signal arrives at it. That you do with all (fictitious) clocks at rest wrt. this frame at any position. In this way you can define locally what "simultaneity" of two events means within this inertial reference frame: Two events at different places are by definition simultaneous when the two synchronized clocks at each of these places show the same time ##t##.
Now since the speed of light by Einstein's 2nd postulate should be the same, independent of the velocity of the source, you get the Lorentz transformations between the space-time coordinates of two different inertial frames, i.e., the frame ##\Sigma'## moving with constant velocity ##\vec{v}=\beta c \vec{e}_1##, which immediately tells you that the synchronized clocks defining the time coordinate ##t## of ##\Sigma## are not synchronized with the synchronized clocks defining the time coordinate ##t'## of ##\Sigma'##. Thus two events being simultaneous wrt. ##\Sigma## are not simultaneous anymore wrt. ##\Sigma'## and vice versa. Since the Lorentz transformations form a group, there can never be contradictions between the description of physical events within either frame of reference. The "coordinate times" ##t## and ##t'## refer to different sets of synchronized clocks.
You can always describe any physical situation in terms of invariant quantities, i.e., scalars, vectors, and (most generally) tensors, which shows that the physics does not depend on the choice of any inertial reference frame.
Usually that's done by choosing some convenient inertial reference frame to define quantities in tensor form. E.g., take relativistic fluid dynamics. There all the needed quantities characterizing the material properties of the fluid are always defined in an inertial frame, where the fluid element is at rest at the time under consideration, like the number density, the temperature, density of thermodynamical potentials, etc. In this way all these densities get scalar fields. In addition you need the four-velocity field ##u^{\mu}## (with ##u_{\mu} u^{\mu}=1##) to express all quantities in an easy way as invariant/covariant quantities. E.g., the four-current of some conserved charge is
$$J^{\mu}(x)=n(x) u^{\mu}(x),$$
where ##n## is the number density as measured in the momentaneous rest-frame of the fluid cell located at ##x##, or the energy-momentum tensor in the case of an ideal fluid
$$T^{\mu \nu}=(e+P) u^{\mu} u^{\nu}-P g^{\mu \nu},$$
where ##e## is the internal-energy density and ##P## the pressure (both as measured in the momentaneous rest frame of the fluid cell). In this way everything is neatly expressed in explicitly invariant tensor quantities (or, as written here, in terms of the corresponding components of all these quantities wrt. to one fixed "observational inertial reference frame", aka the "lab frame").