Reddit mentions of A Journey into Gravity and Spacetime (Scientific American Library)

First, ignore the stretchy fabric picture. It sucks.

I think that before one tries to grapple with the idea of bent (4-dimensional) spacetime, one should first think about the concept of bent (3-dimensional) space, which is not nearly as difficult.

We see examples of bent (2-dimensional) surfaces all the time, like the surface of a basketball or a horse's saddle or the Earth itself. Even if you can't precisely define it, it should make sense that an ant can walk in a "straight" line on the surface of a basketball or a saddle. We call these straight lines geodesics. For example, on a globe, lines of longitude are geodesic but lines of latitude (other than the equator) are not. Seeing this is the first step to understanding what a geodesic is.

An intelligent ant living on a saddle could deduce that he doesn't live in a flat plane (like a tabletop) by making smart measurements of distances and angles using geodesics. (Specifically, on a saddle, two geodesics leaving the same point tend to diverge from each other faster than on a tabletop.) The key point is that the ant can do this without ever leaving the saddle. In other words, even in a two-dimensional universe, you can still tell if your universe is "flat" or "bent." (We prefer the word "curved" rather than "bent.")

Similarly, we can allow for the possibility that our three-dimensional universe is curved. If it's not flat, then our geodesics will not behave the same way as straight lines in Euclidean space. At this point we can imagine a (false) model of the universe in which we live in a curved 3d space, and objects just move along geodesics in the curved 3d space.

Sadly, the reality is harder because it's actually spacetime that's being bent, and the time part behaves differently from the space part in a way that's hard to describe without equations (or at least without understanding special relativity pretty well). Once again, objects move along geodesics, but the big difference here is that the geodesic is now a path through 4d spacetime rather than 3d space. That is, the path itself is tracing out where you are at each time. That's roughly how one can think about what it means to live in a curved 4d spacetime.

(Notice that in my simplified 3d space example, you will trace out the same path no matter how fast you go. In the spacetime setting, two particles pointing the same spatial direction but with different speeds actually point in different "spacetime directions" and will therefore trace out different geodesics in spacetime. I'm only mentioning this because it's relevant to seeing that although the 3d "theory" does make sense, it doesn't make sense as a theory of gravity.)

The last part of the story is the hardest part: The presence of matter causes the 4d spacetime to curve in a certain way. The way it curves is governed by what is called the Einstein Field Equations.

Btw, I remember reading a nice book many years ago by Wheeler. I think it is non-technical in the sense that it's not a textbook filled with equations, but still serious in that it only gives accurate explanations of things, and does have some simple math in it.

i have done a lot of research into this area. people in this thread are a bit shortsighted in my opinion. here are some references that do exactly what you ask and what they state can't be done:

• the einstein theory of relativity: a trip to the fourth dimension by lillian lieber - by a mathematics professor
  
  
• general relativity from a to b by robert geroch - a well known mathematical physicist
  
  
• a journey into gravity and spacetime by john wheeler - that's the advisor of richard feynman and kip thorne and co-author of gravitation by misner, wheeler, thorne
  
  
• discovering relativity for yourself by sam lilley - the author taught relativity to adult students for many decades
  
  
• flat and curved space-times by george f.r. ellis and ruth m. williams - note this book references the books by lilley and geroch in the preface or introduction (can't remember which)
  
  also, the popular science book black hole by marcia bartusiak is a great account of some of the history of general relativity with respect to the acceptance of black holes. none of the books above are popular science though. they describe real physics in real ways.
  
  there are more, but i think this is a good start. have fun!

This change of direction you mention only exists in certain frames of reference, but in the frame of reference of a free-falling object the object feels at rest. This is why astronauts appear to float inside spacecraft, because they’re in freefall, not because they’re far from the Earth.

You can argue that we deal with fictitious forces in pre-relativity mechanics, but Newton’s laws have to do with inertial frames of reference (I’m oversimplifying).

A force is a time-derivative of a momentum vector, and there is even a relativistic version of this concept, but in relativity gravity isn’t a force because......this is hard to explain and trying to learn it was a major motivation for going to grad school and I am still learning. Try getting this book:

A Journey into Gravity and Spacetime (Scientific American Library) https://www.amazon.com/dp/0716750163/ref=cm_sw_r_cp_api_YjJ7Bb866FQ4W