- #36

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The figure below shows four different reference frames:

**A:**The reference frame of someone (inside an elevator, or not) at rest on the surface of the Earth. Whether in an elevator or not, his frame only describes a small region of spacetime.**B:**The reference frame of someone (inside an elevator) in free-fall near the surface of the Earth.**C:**The reference frame of someone inside an accelerating rocket in gravity-free space.**D:**The reference frame of someone (inside an elevator, or not) drifting unaccelerated in gravity-free space.

**A**and

**C**are related to each other via the equivalence principle, and

**B**and

**D**are related to each other via the equivalence principle.

**A**and

**B**are related by a coordinate transformation, as are

**C**and

**D.**

For light-bending, we need to mention three coordinates: ##x, y, t##, where ##y## measures location in the vertical direction, and ##y## measures location in the horizontal direction.

The two different, but equivalent, ways to get to light bending in frame

**A**are:

**Path 1: D to C to A**

- Start with
**D.**Then we know that light initially aimed horizontally will follow the path: ##x_D = c t_D##, ##y_D = y_0##. - Transform to
**C.**Assume that if everything is moving slowly relative to the speed of light, then we can relate the coordinates via: ##t_C \approx t_D##, ##x_C \approx x_D##, ##y_C \approx y_D - \frac{1}{2} g t_D^2##. - This implies that the path of the light will be given by: ##x_C \approx c t_C##, ##y_C \approx y_0 - \frac{1}{2} g t_C^2##
- Go to
**A**using the equivalence principle. So conclude: ##x_A \approx c t_A##, ##y_A \approx y_0 - \frac{1}{2} g t_A^2##

**Path 2: D to B to A**

- Again, start with
**D.**Again we know that light initially aimed horizontally will follow the path: ##x_D = c t_D##, ##y_D = y_0##. - Go to
**B**using the equivalence principle. So conclude: ##x_B = c t_B##, ##y_B = y_0##. - Transform to
**A.**Assume that: ##t_A \approx t_B##, ##x_A \approx x_B##, ##y_A \approx y_B - \frac{1}{2} g t_A^2##. - This implies that the path of the light will be given by: ##x_A \approx c t_A##, ##y_A \approx y_0 - \frac{1}{2} g t_A^2##