Point A to Point B

y = mx + b A straight line is expressed as an equation...Mathematics is how we define our reality....Whether we exist at all is subjective, so then is everything else...http://www.ucl.ac.uk/Mathematics/geomath/level2/fvec/fv31.html

Of course there are straight lines in our universe. A straight line is a mathematical concept that, just like Descartes's famous misused quote, exists because it has been conceived. I doubt that this is what you meant to ask but it is what you actually asked.

From the rest of your question I will assume that you actually meant to ask if anything travels in a straight line trajectory for its entire existence because space has been found to be non-Euclidean due to the effect of black holes and other massive objects that curve space? Depending on how you look at things, everything or nothing moves in a straight line trajectory. Both perspectives are correct. This at first seems like a paradox for we expect that there should be one and only one answer. There can be one and only one answer though if we could discern that there was one and only one frame of reference (perspective) for everything in this universe. We've not found one. So depending on your point of view, someones straight line trajectory is anothers curved trajectory.

No/yes.

Maybe.

A straight line could be considered a segment of a circle with an infinite radius.However, all straight lines follow the curvature of space time,and from the perspective of a photon,always travel in a straight line,even when apparently curved by an observer from another perspective, so I agree with Redfred.

Although it is implied without saying it, a "straight line" usually means "locally straight" and it is represented by the path that light follows in free space, according to some observer. Ideally, it should be an inertial observer, free-falling relative to the masses in the vicinity, but for all practical purposes, we can use light for 'straight' here on earth.

Although the concept may perhaps be extended to "universally straight", i.e. the path that light follows through free space throughout the universe, I'm not sure it has any useful meaning.

"... and eventually, everything will be "curved" into them [black holes]"?

No, I don't think so. Some photons that we observe today traveled for over 13 billion years and I think some of those will make it to - infinity?

-J

Recent observations indicate that Einstein's curvature tensor may be unphysical. The studies of the cosmic microwave background show that the Universe is Euclidean up to the visibility limit. On the other hand gravitational microlensing can be readily seen on astronomical photographs, and astronomical objects follow closed loop orbits. So how it can be explained?

An alternative view is that the quantum fluctuations of the vacuum energy determines the electromagnetic properties of the vacuum, namely the magnetic permeability µ and the dielectric permittivity ε. This is because the fermion-antifermion pairs, which are generated by the energy fluctuations and annihilate within a lifetime comparable to the inverse of their Compton frequency, behave during this short lifetime as dipoles, so the vacuum is polarizable in EM sense. The speed of light is c=1/sqrt(µ×ε), and the vacuum impedance is Z=sqrt(µ/ε). The presence of a massive object modifies the vacuum energy density in its vicinity, modifying in turn µ and ε, and Z and c. So the decrease of vacuum energy density near massive objects causes the apparent spacetime curvature. This may also account for all other relativistic effects.

"Recent observations indicate that Einstein's curvature tensor may be unphysical. The studies of the cosmic microwave background show that the Universe is Euclidean up to the visibility limit."

You have references for such observations?

-J

On the wiki page

<http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation>

the reference 45 at the bottom of the page (but there wer a couple of other articles on arXive, unfortunately I did not save them) reads:

1. ^ de Bernardis, P.; et al. (2000). "A flat Universe from high-resolution maps of the cosmic microwave background radiation". Nature 404 (6781): 955-959. arXiv:astro-ph/0004404. Bibcode 2000Natur.404..955D. doi:10.1038/35010035. PMID 10801117.

"Recent observations indicate that Einstein's curvature tensor may be unphysical. The studies of the cosmic microwave background show that the Universe is Euclidean up to the visibility limit."

I'm afraid that this is not quite what the (some rather outdated, ca. 2000) references claim. The latest information from WMAP has very slight statistical bias towards a closed universe, but a spatially flat, infinite case is not excluded. The largest scales have nothing to do with Einstein's curvature tensor, so I do not know how you have drawn that inference.

Also to be remembered is that a 'flat' (Euclidean) cosmos has to do with space only, not spacetime, which is still curved in the 'flat' case. Even in the case of 'space only', the large scale is not really "Euclidean", because of the expansion of space, so articles calling it that are a bit misleading.

-J

"...a 'flat' (Euclidean) cosmos has to do with space only, not spacetime, which is still curved in the 'flat' case."
Could you explain this or give a reference?

Hi S,

From the Friedmann equations opening assumptions, it is clear that cosmic curvature implies only the spatial part. That's nice and simple, but "flat spacetime" implies Minkowski spacetime, which can at best hold at a very local level. One simply cannot trust Minkowski spacetime at cosmic scales. In a sense, 'the rate of time' must "evolve" in that cosmic energy density evolves and GR teaches us that the rate of time is related to energy density. But, the rate of time evolves relative to what? I can only guess that it is relative to our present rate of time...

-J

Jorrie, I spent a couple of hours looking for those other articles on arXiv I mentioned, but luck was not on my side, so I cannot tell where exactly did I read of the flatness of the space. I hope you won't mind if I address you with this possibly at some later date.

But I'm glad to see that apparently we all from Friedmann on have a lot of trouble with time (as a dimension, as a vector, as a rate of change, as an independent variable in physical functions of time, which, alas, make time itself dependent on those same functions, and otherwise)! How about a thread on that subject on occasion?

Time be as it may, I have quite some trouble to understand the dissociation of time from the space curvature as you mention, especially in view of special relativity, though also in general in view of the general relativity. If a light ray has to follow the curvature of space, and if any observer along the path is supposed to read exactly 'c' from its propagation, then the time should also bend, however rigidly it may try to preserve its proper rate. To me, relativity should apply locally as well as universally, otherwise it would make no sense.

Of course there are people who think relativity is rubbish, and there are equally many who think that vacuum energy fluctuations, in fact the whole quantum mechanics is rubbish. The mere fact that we were not able to seamlessly blend the two into a single theory for a full century speaks of the fact that we have been overlooking a couple of obvious connections between the two.

Best regards!

"If a light ray has to follow the curvature of space, and if any observer along the path is supposed to read exactly 'c' from its propagation, then the time should also bend, however rigidly it may try to preserve its proper rate. To me, relativity should apply locally as well as universally, otherwise it would make no sense."

One possible picture of curved spacetime near a single gravitating mass is shown on right (partial picture).

One can view it as a 'curvature of time' combined with the 'curvature of space'. Along this curve, the local speed of light remains exactly c, but not its speed as measured in the coordinates of a distant observer. This was demonstrated by the Shapiro time delay tests.

Essentially, the slopes of the space component and the time component are the same. The curvatures of the two components are however not the the same.

This is however only a local view and does not apply to the universe at large (e.g. standard cosmic models), where all influences like these are averaged (sort-of). Essentially it is then evenly distributed matter, with empty space between and the models work for that empty space.

Hope it helps.

-J

Jorrie, I just gave you a "good answer" point on that, and I would also give you another one for a "good drawing", however CR4 does not offer any such category yet.

The curvature distinction is now perfectly clear, but that averaging effect on a large scale is not quite so, as far as I can see from the gravitational microlensing of distant objects images by closer galactic clusters. The curvature here is not so extreme as near a black hole, but definitely exists, and IMHO consequently the time should follow a similar amount of bending.

Hi SlowLight: The spacetime curvature around the foreground clusters follow slightly different curves, but the principles are the same as for a BH. As you hinted, there are equal bending effects from space- and time curvature, forming the lensed images. Quasi-Newtonian gravity (with 'corpuscle theory') would have produced exactly half the light-bending, i.e. only the time part. Spatial curvature has no meaning in Newtonian gravity.

As for large scale effects: things like gravitational lensing are due to 'small scale' inhomogeneities in the homogeneous 'toy model' of standard cosmology. A 'flat cosmos' does not mean local flatness anywhere; just flatness on the large scale, on average. This means that the spatial curvature in the voids must be slightly negative and in the 'walls' slight positive.

Does this make sense?

-J

I'm afraid that assumptions do nothing for me. I expect to learn more from the book referred to in this link.

the 'straight line' answer is no.

In the real universe no line can be straight as there is always some 'outside influence'

Will we all disappear into a black hole? Possibly.

But if we do; it would most likely be via a curved trajectory