Leveraging the Eyeball in Relativity: Fast Running Clocks

Leveraging the Eyeball in Relativity: Fast Running Clocks

Joseph West and Nathaniel Shanklin*Dept. of Chemistry and Physics,Indiana State University*Undergraduate

BE06 2017 AAPT Winter Meeting-Atlanta, GA, Feb. 19.

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The “Traveling Twin” ages slowly….so they must see their sibling age “Quickly.” How can that be, if “moving clocks run

slowly?”Consider R. Gruber & R. Price (AJP 1997), acceleration and relative speed are not zero, but aging is the same.Inertial Logs show Lengths are Contracted, but Visual Distances can be Extended. Visual Position.Astronomers

are not inertial observers, and Andromeda is BLUE shifted (γ=1.0005), f = 1.001fo

Moving Clocks

?

are slow?

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A Rectangular spaceship. Headlight effect is more obvious.

Ship moves relative to the Inertial Admiral.The Captain is a Stationary (inertial) Twin.Patti (a Pilot and Traveling Twin) moves around the Ship.Patti will Watch/video the Captain age fast.Why does that Happen according to Patti?

The Setup3

The Setup According to Patti

V=c/2 ϒ = 1.155

Admiral

Patti the Pilot

Captain

FRank

FLuke

BLake

BRooke

All agree that that for t = 0,

FRank

and Patti are at the same location, x = 0, y = 0.

Ship Rest Length is L = 10 ls, Patti says LP = 8.66 ls.

Ship Rest Width is W = 4 ls, Patti WILL say WP = 3.46 ls.

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Given c, and that x =

vt + xo for inertial objects, the “visual position” seen by Patti (xP, yP) is given byyP = y,

The “headlight effect” changes apparent

and

apparent

distance

At t = 0,

BRooke’s

Visual Position is

more

than a

rest length

away as seen (on video) by

Patti

“Visual Position,” the Math

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An Observer at rest (top) and moving towards the white ball (bottom, v=0.9682 c).

The ball appears more to the front and farther away, despite length contraction.C. M.

Savage, A. Searle,and L. McCalman

Am

. J. Phys. 2007, p. 791

“Visual Position,” the Pictures

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When tA

= tFR = 0, the same reading on the Captain’s clock is seen by Patti (distorted) and FRank: tC = - 11.7 sF

Rank expects this from synchronization.Patti (initially) expects this from Slow, poorly synched clocks in the Ship frame, and light travel time

The Inertial Admiral agrees with Patti (for now).

T

wo Pictures of the same Clock

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The Captain’s clock advances by L/v = 20sPatti’s clock advances

by LP/v =17.3 sPatti must have seen the Captain age faster! Remember Andromeda!Nothing new so far…Patti is an inertial observer…so far.

Patti Travels to BRooke

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Patti turns “North” at BRooke, and goes to

BLake.New “Headlight Effect,” and the Captain, momentarily “behind” Patti, is now far in frontBlue Shift dominates Moving Clock, Patti again watches the Captain clock run fast

.The trips to Fluke and FRank repeat the results: Patti sees the Captain aging

fast

.

Patti Turns the Corner

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On

average, Captain ages fast by a factor γThe Admiral log books show thatA) The

Captain ages slowly by a factor γB) Patti ages slowly, on average, by a factor

γ

*

γ (need x, y, velocity addition formulas)

Patti for Students

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The Inertial Admiral’s logs confirm the Captain has a “slow” Moving Clock

Patti sees it approaching, so there is a Blue ShiftThe frequency of Captain’s ticking fC, as seen (on video) by Patti fP, is given by

, tan(

θ

) = y/

xP

θ

is the angle seen by Patti

Logged vs Seen

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When Patti is half way to BRooke.

Visual Position of the Crew

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θ

x

xP

Small time increments for Patti (

dtP) and Captain (dtC) are related by

Attribute factor of

γ

to Moving Clock

Attribute factor of (1-(v/c)cos

θ

) to Blue Shift

Graphs of

fP

are Key to Understanding Fast Clocks

The

time

t

for

the first

leg was evaluated via numerical integration (recognize this?), but only to confirm the known answer...

tC = 20 s, tP = 17.3 s

Accumulated Time, Clock Rate13

The frequency fP,

seen by PattiFast Clocks Using Graphs

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Patti sees the Captain moving towards

her for a long portion of each segment of the round trip (Visual Position and Stellar Abberation).The Blue Shift dominates over the Moving Clock for each segment (but not entire segment).Patti sees, and

expects to see, the Blue Shift Captain in front of her, age quickly.Students can determine value of v required to make The Captain age

fast for the entire observed

trip.

The Fast Aging Twin

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Summer Undergraduate Research Experience (SURE) 2016, Department of Chemistry and Physics, Indiana State

University Aaron Cox and Dr. J. Kinne, Department of Mathematics and Computer Science, Indiana State University.Acknowledgments

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R. P. Gruber and R. H. Price, Am. J. Phys. 1997, p. 979, “Zero time dilation in an accelerating rocket.”

U. Kraus, Eur. J. Phys. 2008, “First-person visualizations of the special and general relativity,” Eq. 1 is for xp.Ron Cowen, Nature (weekly magazine online). May 31, 2012. ““Andromeda on collision course with the Milky Way”V. N.

Matvejev, O. V. Matvejev, and O. Gron, Am. J. Phys. 2016, p. 419. “A relativistic trolley paradox”

C. M. Savage, A. Searle, and L.

McCalman

, Am. J. Phys. 2007, p. 791. “Real time Relativity: Exploratory learning of special relativity” (Images

)

References

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