Notes on Relativity and Cosmology - Physics Department, UCSB

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194 CHAPTER 8. GENERAL RELATIVITY AND CURVED SPACETIME field does not change with time.... but that is a story that we need not go into here 1 . What Einstein needed was a new way of looking at things – a new language in which to discuss gravity that would organize all of this into something relatively simple. Another way to say this is that he needed a better conception of what a gravitational field actually is. This next step was very hard for Albert. It took him several years to learn the appropriate mathematics and to make that mathematics into useful physics. Instead of going through all of the twists and turns in the development of the subject, I’ll try to give you the rough, boiled down version, of how all it all works out. 8.1 A return to geometry You see, Einstein kept coming back to the idea that freely falling observers are like inertial observers – or at least as close as we can get. Recall that, in the presence of a general gravitational field, there really are no global inertial frames. When we talked about our ‘error’ in thinking of a freely falling frame as inertial, it is not the case that there is a better frame which is more inertial than is a freely falling frame. Instead, when gravity is present there are simply no frames of reference that act precisely in the way that global inertial frames act. Anyway, Einstein focussed on the fact that freely falling frames are locally the same as inertial frames. However, he knew that things were tricky for measurements across a finite distance. Consider, for example, the reference frame of a freely falling person. Suppose that this person holds out a rock and releases it. The rock is then also a freely falling object, and the rock is initially at rest with respect to the person. However, the rock need not remain exactly at rest with respect to the person. Suppose, for example, that the rock is released from slightly higher up in the gravitational field. Then, Newton would have said that the gravitational field was weaker higher up, so that the person should accelerate toward the earth faster than does the rock. This means that there is a relative acceleration between the person and the rock, and that the person finds the rock to accelerate away! A spacetime diagram in the person’s reference frame looks like this: 1 If you’re interested, you might look up the difference between ‘stationary’ and ‘static’ spacetimes in a more technical book on General Relativity.
8.1. A RETURN TO GEOMETRY 195 Suppose, on the other hand, that the rock is released to the person’s side. Then, Newton would say that both person and rock accelerate toward the center of the earth. However, this is not in quite the same direction for the person as for the rock: So, again there is a relative acceleration. This time, however, the person finds the rock to accelerate toward her. So, she would draw a spacetime diagram for this experiment as follows: The issue is that we would like to think of the freely falling worldlines as inertial worldlines. That is, we would like to think of them as being ‘straight lines in spacetime.’ However, we see that we are forced to draw them on a spacetime diagram as curved. Now, we can straighten out any one of them by using the reference frame of an observer moving along that worldline. However, this makes the other freely falling worldlines appear curved. How are we to understand this?
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Notes on Relativit
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4 CONTENTS 2.3 Homework Problems .
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6 CONTENTS 8 General Relativity and
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8 CONTENTS
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26 CHAPTER 1. SPACE, TIME, AND NEWT
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Notes on Relativity and Cosmology - Physics Department, UCSB

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