# Could space time curve from the vantage point of earth and out into the universe, and would d = 1 / P in that case give false estimates?

The seventeenth century saw a revolution in astronomy. The invention of the telescope and the acknowledgement of the heliocentric system triggered a race amongst astronomers to measure the parallax of stars - the annual displacement of stellar positions due to Earth's motion around the Sun. In the late 1830s these measurements enabled astronomers to determine the distances to a handful of stars for the first time. From the 1850s onwards, the application of photography to astronomical observations transformed the practice of charting the sky, allowing the compilation of larger and larger catalogues of stellar positions and distances.

That robust marvel of trigonometry applied to observations of other stars, lets me look up into the sky for 6 months, measure the apparent motion, then divide 1 by that, and voila, the distance to the star.

For example, the distance to Alpha Centauri, https://www.wolframalpha.com/input/?i=1+%2F+.77

These distances are used in all textbooks, all databases, all conversations, and does not correct for any space-time curvature whatsoever.

To draw a line on the sky, and use trigonometry, seems almost a bit too simple. Anyone have any thoughts on if space-time might curve in weird ways sort of like how a fish eye lens on Instagram distorts an image?

• I'd suggest that the universe is flatter than the average error on a measurement of parallax for any star. – adrianmcmenamin Dec 13 '16 at 22:39

Yes, but the amount of curvature is too small in scale proportionally to affect such things, especially when one thinks of how away from us the objects are that we observe. Even the moon is a quarter of a million miles away, and the Earth's gravitational well is quite small.

The reason is that when measuring the distance to a star, any star, you are either looking at a star within our own galaxy, or you're doing something wrong. This means that you're not looking at objects far enough away for any possible space-time curvature to be detectable.

That is the reason "simple" trigonometry works, and why these distances are universally used.

We have a very good idea of what causes spacetime to curve. It is described by general relativity, a theory that has been demonstrated to be an excellent model on numerous occasions, both in near and far from the Earth.

Space is curved by mass (or more precisely by mass and momentum as described by the stress-energy tensor), and we know that the space between us and other stars are empty, and so in GR the space is approximately flat.

If spacetime around the Earth was being curved by some unseen mass, the result wouldn't be incorrect distance calculations, instead there would be very obvious distortions: Einstein rings or double images. We don't see this, and so we can be confident that the curvature of spacetime has is negligible when calculating distance by parallax.

• The mass and energy of the milky way itself as a system tough. I've seen schematics of how space time is affected by earth, stars, objects, mass, and it would seem the milky way system as a whole could curve space time. The whole thing could have a gravitational field. – memex Dec 14 '16 at 12:43
• The flux of energy and momentum in the milky way, on the other hand, is not empty space. There are massive amounts of energy which make hundreds of billions of stars orbit the center, with momentum. – memex Dec 14 '16 at 19:46
• My answer stands. – James K Dec 14 '16 at 21:14