In 4-D, the Earth orbits around the sun in a straight line

[emyt]

"in 4-D, the Earth orbits around the sun in a straight line"

someone told me that in 4-D, the Earth actually orbits around the sun in a straight line, could someone try to explain to me how this is so?


thank you

 

[cesiumfrog]

Think of an ant walking in a straight line. If the surface it is walking on is that of an apple, it may well orbit the stem.

 

[pzona]

Are you looking at a fourth spatial dimension or a temporal one?

 

[emyt]

I'm thinking about a fourth spatial dimension. Thanks for the replies but I don't fully understand what cesiumfrog is trying to say.. Wouldn't it be like.. It seems like we are going in a straight line in 3D but we are really going in circles in 4D?

 

[pzona]

The fourth spatial dimension is one of the toughest things to visualize (without using math) because our brains are meant to think in terms of three dimensions. Here's a link to a video which may prove useful in thinking about higher dimensions. I can't comment on the scientific accuracy, since I'm nowhere near good enough with numbers to work these sorts of things out, but it's very interesting either way, and may help indirectly answer your question.

(part one)
(part two)

EDIT: I should probably note that this is mostly about the temporal higher dimensions. The first part is what might help answer your question.

 

[cesiumfrog]

Wouldn't it be like.. It seems like we are going in a straight line in 3D but we are really going in circles in 4D?

Straight lines in 4D, spirals perhaps in ~8D (and obviously still elipses/rosettas when we project our view down into 3D). But, as mathematics actually allows us to study the ant's path just in 2D (without needing to know certain particulars about how the apple's surface is arranged in 3D), likewise for spacetime we don't need to mention ~8D.

The point of the example is that straight lines (like ant that never tries to change direction) can nonetheless still curve back on themselves (if the underlying space is itself intrinsically curved).

 

[diazona]

Personally I think that some of this terminology is a little misleading. When people (who know what they're talking about) say that the Earth travels on a straight line in 4-dimensional space, they really mean a geodesic, which is basically a general term for the closest thing to a straight line. Basically, if you travel through space on what seems to you to be a straight line, you are traveling along a geodesic. If the space is "flat" (undistorted, the way we're used to thinking about it), then a geodesic really is just a straight line, but in a "curved" (distorted) space, a geodesic that seems straight when you're traveling along it can seem curved if you look at it from some distance away - for example, if we were to launch a space probe that could look back on the solar system and see curved orbits.

 

[heldervelez]

The measuring referential must be considered far outside the Solar System: (center of galaxy,.. )
The Sun moves headed to ... (find the direction elsewhere,..)
Earth is attached to Sun, and folows him.
The resultant path is near a straight line (with some ondulatory oscilations).
(The 4D reference on the OP make us try more difficult answers)

 

[mikeph]

Personally I think that some of this terminology is a little misleading. When people (who know what they're talking about) say that the Earth travels on a straight line in 4-dimensional space, they really mean a geodesic, which is basically a general term for the closest thing to a straight line. Basically, if you travel through space on what seems to you to be a straight line, you are traveling along a geodesic. If the space is "flat" (undistorted, the way we're used to thinking about it), then a geodesic really is just a straight line, but in a "curved" (distorted) space, a geodesic that seems straight when you're traveling along it can seem curved if you look at it from some distance away - for example, if we were to launch a space probe that could look back on the solar system and see curved orbits.

This.

Think about a lower dimensional analogy. Does a straight line exist in 3D space whose projection onto a 2D plane gives a circular line?
How the projection of a straight line in 4D space onto a 3D volume possibly be a curved line?
In terms of geometry and topology, I think this is nonsense. It only makes sense in that the line is a geodesic, ie. the result of an extremisation of the action of some metric.. (I think this is the right phrasing, it has been a year since I did GR)

 

[DaveC426913]

This.


consider a straight line in 3D space, the projection of which onto a 2D plane is a circle

Your 3D space is too general. Look a the specific case of a closed 3D object.

Consider a world line on the surface of the apple (any cut that passes through the core). It deviates neither left nor right. (It might be easier to visualize if you grid the apple.) When projected into 2D dimensions, it will be a straight line across the plane (which will eventually loop back on itself).

 

[mikeph]

This is the opposite way round to OP's question- he is saying that a curved line in n-dimensional space is straight in n+1 dimensions. The projection of a circular loop on 3D space can become a straight line if projected onto the right plane, I agree, but not the other way around...

 

[cesiumfrog]

How [can] the projection of a straight line in 4D space onto a 3D volume possibly be a curved line?
In terms of geometry and topology, I think this is nonsense. [...] The projection of a circular loop on 3D space can become a straight line if projected onto the right plane, I agree, but not the other way around...

I think you're breaking the analogy, by thinking of projections specifically from Euclidean 3D to a 2D plane. In GR, the 4D space is curved/non-Euclidean. (We could embed it in flat space given plenty of additional dimensions, but because the geodesics are constrained to a 4D hyperslice, they are not straight lines in the higher flat space.)

Using the curved 2D surface of the apple as the analogy to GR's 4D (and so deliberately ignoring that the apple's surface happens to be embedded in Euclidean 3D), the analogy to your 4D->3D projection is a projection to one arbitrary 1D coordinate defined across the surface (not defined in reference to the 3D embedding space). In particular, both in this example and in GR, you will now be able to find open spirals (in spacetime or the apple surface, repectively) that are projected to closed loops (in 3D or the number line, respectively). Does this make sense to you, without me fleshing out the explanation by a specific choice of coordinate for projection to 1D?

You can still argue that geodesics on a curved manifold are not really straight, but obviously they are exactly the straightest paths possible on intrinsically curved manifolds. Nobody argues that the path between two cities "as a crow flies" can't be called straight (despite that technically it isn't, because the Earth is round, and the true straight path would be a tunnel that departs the Earth's surface entirely), and likewise I think it would be overly pedantic to deny the intuitive notion of a ant marching neither left nor right but (locally) "straight".

 

[mikeph]

GR is 4D in the sense that you have 3 space and 1 time coordinates, OP specifically stated there were 4 spatial dimensions, very different situations. OP also never said the space was curved so I am not making this leap forward without a reason to believe it needs to be. I know that a curved n-1 dimension hyperplane in n dimensional space can contain a curved line in n dimensions, but a straight line in n-1 dimensions along that hyperplane.. but I don't see why we are talking about curvature.An open spiral can be projected to make a circle, but I don't see why that matters, we are looking for straight lines that project to circles, and a spiral is not straight. Maybe the solution is that you are projecting a straight line onto a curved space? But what type of curved space do you need for a straight line to be projected onto a circle? I don't see how it can happen if the space is continuous and connected.I have no problem with the geodesic explanation, there is no real sense of straight line along a curved manifold, only geodesics. But I don't think that is what OP is asking, I think the answer to his question is simply "no".

 

[A.T.]

But I don't think that is what OP is asking,

Look:

someone told me that in 4-D, the Earth actually orbits around the sun in a straight line, could someone try to explain to me how this is so?

Someone told him something, and he tries to reproduce it. It is not difficult to guess what that someone meant, even if that someone didn't express himself correctly or emyt failed to reproduce it exactly.

OP also never said the space was curved

Neither did he say it was flat.

OP specifically stated there were 4 spatial dimensions,

So what? A geodesic in curved 4D-space can be a circle when projected onto 2D-space.

I think the answer to his question is simply "no".

I think otherwise.

 

[mikeph]

I am getting more and more confused, I think we are talking about different things,

To me it is interesting to think about the flat space and if you can project a straight line onto a curve, nobody has responded to this.

I know a geodesic can be curved when projected, that's not what I am talking about. For me that point was resolved at post #7 which I agreed with.

 

[emyt]

I see that this thread is still going, I verified (by looking it up) that "going in a straight line" is really going as a geodesic. Sorry for the confusion but this stuff is out of my depth, I wasn't entirely sure what the person was talking about, that's why I posted it here

thanks for the replies :)