If gravity is a "curvature of space" rather than a force, why do a ball and bullet follow different paths?
If gravity can be defined as the curvature of space rather than a force of attraction, why does not a bullet shot out of a gun, say perpendicular to the Earth's crust, and a ball thrown by me on the same trajectory (but obviously at a much slower speed) follow the same curve? If gravity is actually curved space and if falling objects are simply following the natural curves of space why does each object have its own curve?
This is a great question which goes to the heart of why Einstein said gravity is the curvature of space-time, rather than just the curvature of space.
Consider the diagram below, which shows the situation you describe in your question. The ball and the bullet both start off along the same path in space (i.e. they both start off moving horizontally away from the person's head). However, as we know, their paths quickly diverge -- the bullet will travel much farther before hitting the ground than the ball does.
This might not seem strange at first, but under Einstein's reconception of gravity, it's a major problem! Einstein's idea (discussed further on our relativity page) was that there is no such thing as a "force" of gravity which pulls things to the Earth; rather, the curved paths that falling objects appear to take are an illusion brought on by our inability to perceive the underlying curvature of the space we live in. The objects themselves are just moving in straight lines.
If this is true, however, then the ball and the bullet which start off on the same path should logically continue on the same path. After all, if you imagine walking on a curved surface such as the Earth, if you start off walking in a straight line towards the east and your friend starts from the same location running in a straight line towards the east, you'll both follow the exact same path! It doesn't matter how fast you're going; you'll both (eventually) reach the same location. So why don't the ball and the bullet wind up in the same location too?
The only way to get around this problem in Einstein's theory is to say that it is not just "space" which is curved; rather, it is "space-time." To understand this, it's helpful to look at the illustration below. This illustration shows the same ball and bullet as above, only now they are plotted on a diagram of space AND time together ("space-time").
The horizontal axis is the same as it was before; it represents distance in the left and right direction. The vertical axis, however, no longer represents distance in the up and down direction like it did before; instead, the vertical axis represents time, or, specifically, how much time has elapsed since the ball and bullet were released. Whereas the first diagram was a picture of something you could imagine seeing with your eyes (and therefore I included a person in the picture), the second diagram is something you can only imagine in your head. What the diagram tells us, however, is that at any point in time, the bullet has moved through more space than the ball has, which makes sense since the bullet is moving at a faster speed.
The above diagrams show that although the ball and the bullet start off along the same direction in space, they actually start off along different directions in space-time. So if we agree that space-time, and not space, is the proper arena in which to consider the question, then we can understand why the ball and the bullet don't wind up in the same place at the end of their trip. Just like it's not surprising that if you start off walking east and your friend starts off walking northeast, you will end up in different places, it also isn't surprising that the ball and the bullet end up in different places, since they started off in different directions!
Furthermore, let's consider what happens if two objects start off along the same path in space-time. You should be able to convince yourself that the only way this can happen is if the two objects start off in the same direction and at the same speed. For example, we might fire two different bullets from the same gun, with each bullet made of a different material. Or we might fire a bullet and a cannonball from the same point in space, with each starting out at the same speed. It turns out that in these cases, if no other forces such as air resistance act on the objects being fired, they will follow the exact same paths and hit the ground at the exact same time and exact same place. This is completely consistent with Einstein's theory -- the path that an object takes through space-time doesn't depend on the mass of the object or on the material it is made of; it only depends on the initial direction that the object starts off in.
In some sense, therefore, what Einstein's theory tells us is that we really need to consider space-time, rather than space, as the fundamental "playing field" upon which the events of the universe occur. It is a profound realization to understand this fact -- all the objects around us actually exist in a realm of "space-time" that is much more complex than the simple realm of space in which we perceive them.
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