Is Spacetime Linear? - The Universe in a Nutshell

[Justinmcg67]

I was wondering about this question when I was reading my new book The Universe in a Nutshell by Stephen Hawking and the images in the book depict space time as a linear fabric. I don't understand how this could be if the sun has a spherical gravitational pull, as we see at the very outermost reaches of our solar system. So if space time is curved down on a linear fabric, how are objects gravitattionally bound below and on top of the Sun?

I'm sure there is an explanation, I however can not seem to find it. So any light shed on this subject is very much appreciated.

 

[Muphrid]

What do you mean by a "linear" fabric? Functions can be linear. Operators can be linear.

The fabric metaphor is an easy way for people to understand spacetime curvature by analogy to stretching. It's not the most precise metaphor, though. Instead of the single sheet curving downwards due to gravity, imagine instead a net lying flat on a table. Pretend this net can be stretched any way you like without breaking. This net represents spacetime, and any stretching gives rise to "curvature".

To complete the metaphor, imagine instead of a flat net you have a grid of connected ropes in 3d space which can stretch like the net.

 

[Justinmcg67]

The 3D net makes a lot more sense to be honest. And by linear I mean flat, straight, like a line. Thank you for the answer though as it does shed some light on the topic for me.

 

[crissyb1988]

Just repeating the above, the metaphor isn't quite exact. Space time ISNT a 2D "fabric" that curves in the presence of mass. We must add another dimension so it is now a 3D "fabric" that curves in the presence of mass. For simplicity diagrams show a 2D fabric because a 3D fabric would be too complicated.

 

[pmacias]

I can provide some insight into this question. The concept of spacetime being linear or curved is a fundamental aspect of Einstein's theory of general relativity. According to this theory, spacetime is a four-dimensional fabric that is curved by the presence of mass and energy. This curvature is what we experience as gravity.

The images in Stephen Hawking's book may depict spacetime as a linear fabric, but it is important to understand that this is simply a visual representation. In reality, spacetime is a complex and dynamic structure that is constantly changing and being influenced by the presence of matter and energy.

In regards to the specific example of the sun's gravitational pull, it is important to note that the sun's mass does indeed curve spacetime in a spherical manner. However, this curvature is not limited to just the space directly above and below the sun. It extends throughout the entire solar system, creating a gravitational field that affects all objects within it.

In fact, it is this curvature of spacetime that allows objects to be gravitationally bound to the sun. The sun's mass creates a deep well in the fabric of spacetime, and objects with less mass, such as planets, are pulled towards this well and orbit around the sun. This is similar to how a bowling ball placed on a trampoline will create a dip in the fabric, causing smaller objects, like marbles, to roll towards it.

In summary, while the concept of spacetime being linear may help us visualize and understand certain aspects of general relativity, it is important to remember that it is a much more complex and dynamic structure. The curvature of spacetime is what allows for the existence of gravity and the binding of objects in our solar system and beyond. I hope this helps shed some light on the subject.