Twin Paradox in General Relativity

A more detailed description of the Twin Paradox in Albert Einstein's General Theory of Relativity.

[EugeneKhutoryansky]

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[EugeneKhutoryansky]

Everyone, if you liked this video, you can help more people find it in their search engine by clicking the like button and writing a comment.  Also, if you haven't already seen it, two days ago I uploaded a video on gravity's effect on the flow of time.  Thanks.

[mikegale]

No need to invoke GR to break the symmetry. The older twin sees more redshifted light from the younger one. If you can use Matlab or Octave, try this:
function ageDiff = TwinParadox(v2, v1=0, vo=0)
% SYNOPSIS: Plot worldlines and time ticks for the Twin Paradox scenario
% v2 is velocity of the 2nd twin in the CoM frame
% v1 is velocity of the 1st twin in the CoM frame
% vo is the velocity of the observer in the CoM frame


v = [vo v1 v2];
worldline{1} = zigzag(0, v(1));
[ worldline{2} age1 red1 white1 blue1] = zigzag(v(2), v(1));
[ worldline{3} age2 red2 white2 blue2] = zigzag(v(3), v(1));

% Minkowski grid lines
xm2 = linspace(-1, 1, 201);
tm2 = [];
for i=0:11
tm2 = [tm2; sqrt((i/10)^2+xm2.*xm2)];
endfor
xm3 = tm2(:, 101:201);
tm3 = xm2(101:201);
xm1 = -xm3;
tm1 = tm3;


% Lines of sight
v(1) = -v(1);
for i=2:3
v(i) = (v(i)+v(1))/(1+v(i)*v(1));
endfor
los = {};
for i=1:3
los{i} = [[0 0]; [sign(v(i)) abs(v(i))]];
endfor


% Plot results
figure, hold on, grid on
ageDiff = age2 - age1;
title (['Twin Paradox (\DeltaT = ', num2str(ageDiff), ')'])
xlabel ('Space')
ylabel ('Time')
xlim([-1 1])
ylim([0 1.5])
color = 'kmg';
names = {'CoM'; 'Twin #1'; 'Twin #2'};
for i=1:3
plot(worldline{i}(:, 2), worldline{i}(:, 1),
[ color(i) ';' names{i} ' (v = ' num2str(v(i)) ');'],
'linewidth', 2);
endfor
for i=1:11
plot(xm1(i, :), tm1, 'k-.')
plot(xm2, tm2(i, :), 'k-.')
plot(xm3(i, :), tm3, 'k-.')
endfor
for i=1:length(red1)
plot(red1{i}(:, 2), red1{i}(:, 1), 'r')
endfor
for i=1:length(red2)
plot(red2{i}(:, 2), red2{i}(:, 1), 'r')
endfor
for i=1:length(blue1)
plot(blue1{i}(:, 2), blue1{i}(:, 1), 'b')
endfor
for i=1:length(blue2)
plot(blue2{i}(:, 2), blue2{i}(:, 1), 'b')
endfor
for i=1:3
data = los{i};
plot(data(:, 1), data(:, 2), color(i), 'linewidth', 2);
endfor
plot(white1(:, 2), white1(:, 1), 'k')
plot(white2(:, 2), white2(:, 1), 'k')
endfunction


function [ worldline age red white blue] = zigzag(v, vo)
g = 1 / sqrt(1 - v^2);
t = g * [0:0.1:1];
x = v * t;
secondleg = find(t > 0.5);
x(secondleg) = v * (1 - t(secondleg));
roundtrip = find(sign(v) * x >= 0);
ticks = [t(roundtrip)' x(roundtrip)'];
worldline = [[0 0]; [0.5 v * 0.5]; [1 0]];


firstleg = find(ticks(:, 1) <= 0.5);
red = {};
for i=1:length(firstleg)
j = firstleg(i);
red{i} = [[ticks(j, 1) ticks(j, 2)]; [ticks(j, 1) + abs(ticks(j, 2)) 0]];
red{i} = boost(red{i}, vo);
endfor


secondleg = find(ticks(:, 1) > 0.5);
blue = {};
for i=1:length(secondleg);
j = secondleg(i);
blue{i} = [[ticks(j, 1) ticks(j, 2)]; [ticks(j, 1) + abs(ticks(j, 2)) 0]];
blue{i} = boost(blue{i}, vo);
endfor


white = [0.5, v/2];
white = [white; [0.5 + abs(v)/2, 0]];
white = boost(white, vo);


age = 1 / g;
worldline = boost(worldline, vo);
endfunction


function [result] = boost(data, v)
g = 1 / sqrt(1 - v^2);
t = g * (data(:, 1) - v * data(:, 2));
x = g * (data(:, 2) - v * data(:, 1));
result = cat(2, t, x);
endfunction

[EugeneKhutoryansky]

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Thanks.

[saragct]

This is the only best explanation of twin paradox that I have ever read. This topic had always troubled me but not anymore. Thank you!

[wills]

First video I have seen on this subject which comes close to resolving the twins paradox.
Acceleration and gravity are indistinguishable when it comes to the gravity effects on time.

One exception, however. The distance between clocks does not matter. You could accelerate Adam into orbit around the earth, bring Adam back to earth and Sarah and Adam would also be a different age. This experiment was done with the shuttle.

[TinaGong]

I love your videos! Also, the music includes some of my classical favorites :)

[hamidghorbanbeigi]

Thank you! I had this question in my mind for so long and you explained it in such an elegant way

[bartkwezelstaart]

1812 Overture. A great physics youtuber and a great taste of music... Is this going to get any better?

[vinnv]

This is shaping up to be one of my favorite channels. A love seeing new videos of yours in my feed. Great video, as always.

[NeedsEvidence]

Nice video. Note that, since special relativity is a self-contained theory, it's also possible to resolve the twin paradox without reference to general relativity by invoking *relativity of simultaneity* in the context of the frame change experienced by the twin at the return point at large distance from earth.

[Alex-]

Great video, but I'm still confused about what is different for the two people. From Adam's point of view (when he accelerates) he experiences a gravitational field for the entire universe (which cancels out with his engine thrust) which results in Sarahs clock moving really fast compared to his clock. If this is correct couldn't you also say that from Sarah's point of view a gravitational field is applied on Adam's ship which makes his clock move really fast as well...

[spirit-teacher]

Eugene, you are the best teacher of theoretical physics in YouTube. Hands down.

[mau_lopez]

Absolutely great video ! (as many many others from Eugene). After years of looking for a clear, comprehensive and simple explanation of the Twins Paradox, finally, for me this has been the one! Thanks a lot Eugene !

[nathanos]

We need to address this problem without involving gravity and acceleration. Assume instant acceleration so that 100%of journey is spent in constant speed. In this case it will be impossible to solve the paradox with just the two observers. They both need to be measured relative to a third reference frame, that being the universe itself. Only one of them is moving faster than the other relative to the whole universe.

[jianweisatasata]

Why isn't it such that Adam is actually stationary and the Earth is the one accelerating back to Adam and hence won't the aging of Sarah being observed by Adam, also be observed on Adam by Sarah? It seems you sort of establish the fact that it is Adam that is moving and accelerating and not Sarah. But isn't it relative? Like if Sarah is on a rocket ship instead of the Earth?

[vincemarenger]

FINALLY. For EVERY twin paradox explanation, I was always left with the same freaking question :
“Well, from the traveler's point of view, he is standing still and the earth is the one accelerating mid journey. So how does he know he’s the one accelerating ?” NOW I understand !
THANK YOU 😊

[starwarsjk]

1 observer observes special relativity, while the other observes general relativity. That sounds fun xD

[blacktigershearthstoneadve]

So, let me summarime it. Suppose two ship fly in different directions, then turn back at the "same" time or not, does not matter (it is relative anyway). So whoever's ship applied less force with it's engines will be older when they meet.

[wenniezhao]

The twin paradox is seemly solved by considering the turning-around trip. However, let us pause for a second. The way to solve the twin paradox doesn't solve the time dilation paradox itself at all. Let's use two atomic clocks, one on the earth and the other one on the spaceship traveling at half speed of light. From my perspective, the clock on the spaceship runs slowly, r = 1.15. So at the noon of the first day on spaceship, the clock on spaceship shows the time 12:00 and the one on the earth shows the time 13:48. Now from the astronauts’ perspective, the clock on the earth runs slowly with the same value of time dilation. So when the clock on spaceship shows the time 12:00, the one on earth shows the time 10:26. So how can a clock show two different times at the same spaceship time?

To solve this time dilation paradox, I think we have to use Einstein's relativity of simultaneity. By using the spaceship space-time frame, the two events, the clock on the spaceship showing12:00 and the clock on earth showing 10:26, happened at the same time. However, by using the earth space-time frame, these two events no long happened at the same time. Because of the spaceship is travelling away from the earth, so the clock on earth showing 10:26 happened earlier. So when the clock on space shows 12:00, the clock on earth shows a time later than 10:26, that is 13:48. Likewise for the two events happened at the time by using the earth space-time frame, the clock on the space showing 12:00 and the clock on earth showing 13:48, also no long happened at the same time by using the spaceship space-time frame. Since the spaceship is travelling away from the earth, the clock on earth showing 13:48 happened later. So when the clock on spaceship shows12:00, the clock on earth shows the time earlier than 13:48, that is 10:26. So now the time dilation paradox is solved :)