- #1

astrodummy

- 16

- 11

- TL;DR Summary
- I refer to a project printed in Taylor and Wheeler's "Spacetime Physics" on pages 186-187. It was discussed here a while back but no satisfying solution was offered. I offer one here.

This post builds on a previously closed thread here:

https://www.physicsforums.com/threads/lightbulb-paradox.141191/#post-1145674

I will not describe the problem here without the copyright owner's permission. I, like the OP in the original thread, am keen to see if anyone else who has this text has solved the problem, since no solution is offered in the text.

My solution is to determine the spacetime interval between the power being disrupted to the bulb and the power being restored to the bulb in each reference frame. If these are the same then there is no paradox and special relativity holds.

NOTE: I mistakenly used 2 metres as the distance between the lamp and the start of gap instead of 4 meters in the rail frame and 1 meter instead of 2 in the slider frame. This does not affect the spacetime interval, just some of the event coordinates.

Event coordinates were determined used a spacetime diagrams (attached) drawn to scale and plotting the various world lines.

RAIL FRAME Event Coordinates

Working out the event coordinates in the rail frame is pretty easy. First plot the world lines of bar A and B on the slider as it traverses the gap in the top rail. The power lost world line starts as soon as the slider enters the gap and the power on world line starts when bar B reaches the end of the gap in the top rail. From there these world lines travel toward the left as light-like signals until they intersect with spatial line x=-2, where the bulb is. These intersections are easy enough to see on the spacetime diagram.

Event #1: Bar A enters gap

##(0,0)##

Event #2: Bar B exits gap

Time to travel 1 metre when ##\gamma =2~is~\frac {2} {\sqrt{3}} m##

##(2,\frac {2} {\sqrt{3}} )##

Event #3: Power loss hits bulb

##(-2,2)##

Event #4: Power restored to bulb

##(-2,\frac {2}{\sqrt{3}}+4)##

So we are interested in the spacetime interval between Event #3 and Event #4.

This interval, as we know is given by ##interval^2 = time^2 - space^2## which is

##interval^2 = (\frac {2} {\sqrt{3}}+4 - 2)^2 - ( -2 - -2)##

##interval^2 = (\frac {2}{\sqrt{3}}+2)^2 - 0##

##interval = \frac {2} {\sqrt{3}}+2## m

SLIDER FRAME Event Coordinates

Things get a little more involved in the slider frame since both the rail and the power off and on signals travel to the left relative to the at rest slider. I had to derive the linear equations for the power off and power on world lines, then determine the intersections with the bulb world line in order to arrive at the event coordinates #3 and #4.

Linear equations

Bulb worldline

##ct=-\frac{2}{\sqrt{3}}(x+1)~~~##[1]

Power off worldline

##ct=-x~~~##[2]

Power on worldline

##ct=-x-\frac{2}{\sqrt{3}}+2~~~## [3]

Find intersection of line [1] and [2] to derive Event #3 coordinates

##-x = - \frac {2}{\sqrt{3}}(x+1)##

## x= \frac {2}{\sqrt{3}-2}##

## ct=\frac {-2}{\sqrt{3}-2}##

Find intersection of line [1] and [3] to derive Event #4 coordinates

##-x+\frac {\sqrt{3}} {2} = \frac {-2} {\sqrt{3}}(x+1)##

##x = -6 - 4\sqrt{3}##

##ct = 8 + \frac {10} {\sqrt{3}}##

Event #1: Bar A enters gap

##(0,0)##

Event #2: Bar B exits gap ( BEFORE Event #1! )

##(2,\frac {\sqrt{3}} {2})##

Event #3: Power loss hits bulb

##(\frac {2} {\sqrt{3}-2},\frac {-2}{\sqrt{3}-2})##

Event #4: Power restored to bulb

##(-6-4 \sqrt{3},8+\frac {10}{\sqrt{3}})##

Now at last we can determine the spacetime interval between Event #3 and Event #4.

This interval, as we know is given by ##interval^2 = time^2 - space^2## which is

##interval^2 = (8+\frac{10}{\sqrt{3}}+\frac {2}{\sqrt{3}-2})^2 - (-6 -4\sqrt{3} - \frac {2} {\sqrt{3}-2})^2##

##interval^2 = (4+\frac{4}{\sqrt{3}})^2 - (-2(1+\sqrt{3}))^2##

##interval^2 = \frac{32}{3}(2+\sqrt{3}) - 8(2+\sqrt{3})##

##interval^2 = \frac{8}{3}(2+\sqrt{3})##

##interval = \frac{2}{\sqrt{3}}+2## m

CONCLUSION

Since the spacetime interval between Events #3 and #4, the bulb flickering off and back on, is the same in each reference frame there is no paradox. Special Relativity still works!

https://www.physicsforums.com/threads/lightbulb-paradox.141191/#post-1145674

I will not describe the problem here without the copyright owner's permission. I, like the OP in the original thread, am keen to see if anyone else who has this text has solved the problem, since no solution is offered in the text.

My solution is to determine the spacetime interval between the power being disrupted to the bulb and the power being restored to the bulb in each reference frame. If these are the same then there is no paradox and special relativity holds.

NOTE: I mistakenly used 2 metres as the distance between the lamp and the start of gap instead of 4 meters in the rail frame and 1 meter instead of 2 in the slider frame. This does not affect the spacetime interval, just some of the event coordinates.

Event coordinates were determined used a spacetime diagrams (attached) drawn to scale and plotting the various world lines.

RAIL FRAME Event Coordinates

Working out the event coordinates in the rail frame is pretty easy. First plot the world lines of bar A and B on the slider as it traverses the gap in the top rail. The power lost world line starts as soon as the slider enters the gap and the power on world line starts when bar B reaches the end of the gap in the top rail. From there these world lines travel toward the left as light-like signals until they intersect with spatial line x=-2, where the bulb is. These intersections are easy enough to see on the spacetime diagram.

Event #1: Bar A enters gap

##(0,0)##

Event #2: Bar B exits gap

Time to travel 1 metre when ##\gamma =2~is~\frac {2} {\sqrt{3}} m##

##(2,\frac {2} {\sqrt{3}} )##

Event #3: Power loss hits bulb

##(-2,2)##

Event #4: Power restored to bulb

##(-2,\frac {2}{\sqrt{3}}+4)##

So we are interested in the spacetime interval between Event #3 and Event #4.

This interval, as we know is given by ##interval^2 = time^2 - space^2## which is

##interval^2 = (\frac {2} {\sqrt{3}}+4 - 2)^2 - ( -2 - -2)##

##interval^2 = (\frac {2}{\sqrt{3}}+2)^2 - 0##

##interval = \frac {2} {\sqrt{3}}+2## m

SLIDER FRAME Event Coordinates

Things get a little more involved in the slider frame since both the rail and the power off and on signals travel to the left relative to the at rest slider. I had to derive the linear equations for the power off and power on world lines, then determine the intersections with the bulb world line in order to arrive at the event coordinates #3 and #4.

Linear equations

Bulb worldline

##ct=-\frac{2}{\sqrt{3}}(x+1)~~~##[1]

Power off worldline

##ct=-x~~~##[2]

Power on worldline

##ct=-x-\frac{2}{\sqrt{3}}+2~~~## [3]

Find intersection of line [1] and [2] to derive Event #3 coordinates

##-x = - \frac {2}{\sqrt{3}}(x+1)##

## x= \frac {2}{\sqrt{3}-2}##

## ct=\frac {-2}{\sqrt{3}-2}##

Find intersection of line [1] and [3] to derive Event #4 coordinates

##-x+\frac {\sqrt{3}} {2} = \frac {-2} {\sqrt{3}}(x+1)##

##x = -6 - 4\sqrt{3}##

##ct = 8 + \frac {10} {\sqrt{3}}##

Event #1: Bar A enters gap

##(0,0)##

Event #2: Bar B exits gap ( BEFORE Event #1! )

##(2,\frac {\sqrt{3}} {2})##

Event #3: Power loss hits bulb

##(\frac {2} {\sqrt{3}-2},\frac {-2}{\sqrt{3}-2})##

Event #4: Power restored to bulb

##(-6-4 \sqrt{3},8+\frac {10}{\sqrt{3}})##

Now at last we can determine the spacetime interval between Event #3 and Event #4.

This interval, as we know is given by ##interval^2 = time^2 - space^2## which is

##interval^2 = (8+\frac{10}{\sqrt{3}}+\frac {2}{\sqrt{3}-2})^2 - (-6 -4\sqrt{3} - \frac {2} {\sqrt{3}-2})^2##

##interval^2 = (4+\frac{4}{\sqrt{3}})^2 - (-2(1+\sqrt{3}))^2##

##interval^2 = \frac{32}{3}(2+\sqrt{3}) - 8(2+\sqrt{3})##

##interval^2 = \frac{8}{3}(2+\sqrt{3})##

##interval = \frac{2}{\sqrt{3}}+2## m

CONCLUSION

Since the spacetime interval between Events #3 and #4, the bulb flickering off and back on, is the same in each reference frame there is no paradox. Special Relativity still works!