Relationship between spacetime, acceleration, and gravity?

[uchihajeff]

I've been reading "The Universe in a Nutshell" and there is one thing I don't understand very well. What is the relationship between spacetime, acceleration, and gravity?

Please explain to a level of my understanding: I'm in 9th grade.

More specifically, what I don't get is why the "equivalence between acceleration and gravity" would work if spacetime was curved (or why it wouldn't if spacetime wasn't curved).

If my question isn't clear enough, please tell me.

 

[dextercioby]

I've been reading "The Universe in a Nutshell" and there is one thing I don't understand very well. What is the relationship between spacetime, acceleration, and gravity?
Please explain to a level of my understanding: I'm in 9th grade.
More specifically, what I don't get is why the "equivalence between acceleration and gravity" would work if spacetime was curved (or why it wouldn't if spacetime wasn't curved).
If my question isn't clear enough, please tell me.

9-th grade,u say.That would mean 15-16 yrs old.Maybe a little too 'fresh' for books by Stephen Hawking...

Anyway:the strong principle of equivalence formulated by Albert Einstein in November 1915 (i don't know in which article exactly,11/18/25th,one of them...) requires that the spactime continuum be curved.The Special Theory of Relativity (oficially 10 years older that its generalization) explains relativistic dynamics of bodies that are not accelerated.That is to say dynamics in inertial reference frames.This is done in a 4D flat space called "Minkowski space".Once one considers general coordinate transformations to include noninertial reference frames and accelerated bodies,the spacetime automatically curves.The strong principle of equivalence places gravity and accelerated reference frames on equal footing,so gravity can be decribed only in curved space.

I hope that in the 9-th grade u know what reference frames are and the difference between inertial and noninertial ones.If u don't,then learn that 2 reference frames that are moving so that one has a constant velocity "v" (means acceleration '0') wrt to the other are said to be 'inertial'.Two reference systems that are moving so that one has a nonzero acceleration "a" wrt the other are called "noninertial reference frames".

Understanding the concepts of 'flat' and 'curved' spaces is more difficult though.To your knowledge,the 2D plane (in which u can pick 2 the cartesian coordinates Oxy) is an example of a 2D 'flat' space.Minkowski space is flat and has 4 dimensions.The surface of a sphere is an example of 2D 'curved' space.The 'environment' for the GTR (the 'spacetime') is an example of 4D curved space.

Daniel.

 

[NateTG]

More specifically, what I don't get is why the "equivalence between acceleration and gravity" would work if spacetime was curved (or why it wouldn't if spacetime wasn't curved).

Ultimately the problem has to do with we consider to be straight lines. Newton, and I am sure others, observed that undisturbed objects travel in 'straight lines'. From a Newtonian perspective, objects in a gravitational field do not travel in straight lines due to the force of gravity acting on them, and it's all ok.

Let's digress for a moment, and imagine a rifleman in a spinning bowl. The rifleman, sitting in the bowl, shoots out a tracer bullet. From the rifleman's perspective, the bullet travels in a spiral, but, from his past experience he knows that the bullet essentially goes in a straight path. Now, the rifleman can either conclude that there is some force acting on the bullet, or that this spiral path is indeed a straight line. Now, you might (from a Newtonian) perspective, suggest that the Rifleman is in an accelerated reference frame, and that Newtonian mechanics is not valid in accelerated reference frames. However, there does not appear to be any test that can distinguish between gravity and an accelerated reference frame. A force that has the property that it is indistinguishable from an accelerated reference frame is called an 'inertial force'. (Literally, inertial essentially means that the force doesn't do anything.)

From the GR perspective, there are no inertial forces, but instead, space is bent so that the 'straight lines' that undisturbed objects travel along are not necessarily what you would normally consider to be straight lines. It's worth noting that objects traveling at different speeds in a gravitational field are traveling along different paths, so the straight lines have to be different for different speeds. Consequently, it's necessary to have bent space time rather than just bent space.

From a scientific perspective, this notion of bent space time would be considered a curiosity, except that it makes predictions that differ from the predictions made by Newton, and the predictions based on the bent space time notion are more accurate than those made by Newtonian mechanics. Since Newtonian mechanics is already very accurate, and generally simpler to use, it is still quite popular, but when extreme precision is necessary, GR has to be used.

 

[Tide]

Imagine that you were suddenly transported into a closed chamber with no access to the outside world. Einstein's equivalence principle basically says that you will be unable to distinguish between whether your chamber is sitting on the surface of a planet (subject to its gravitational force [curved space]) or your chamber is accelerating through empty space.