Is spacetime linear?

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 isnt 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.