I have been wondering exactly what goes on when we observe light from a distant galaxy, or when we observe Cosmic Microwave Background (CMB) radiation.
Intuitively, it just seems to me that at the early stages of the universe, the distances between things would be small enough that it wouldn't take much time for light to go between any two objects that existed. So it seems to me that any light that originated in the early universe would have reached Earth (or the center of mass of all the stable particles that are part of our solar system today, wherever they all were then) long ago.
I would like someone to follow my analysis and tell me where I am going wrong. Since I don't know how to produce diagrams on this forum, I will spell out my analysis step by step, and hopefully you will follow along with paper and pencil.
1. Draw a big square on a piece of paper. I don't think you need to use a ruler or be very accurate.
2. Starting at the lower left, going counterclockwise, label the corners B, R, C, and E
3. Draw in the diagonals and label their intersection M
4. Let BE represent a time axis with B representing the time of the Big Bang, and E representing the time on Earth now.
5. Assume the age of the universe is 14.43 billion years old (I think that is close and it makes the math work out nicely). That establishes the scale on the time axis. You can write 'time' and draw an arrow pointing up just to the left of your square to label this axis.
6. Let the line BR represent a dimension of space that lies in the direction of an observation made from Earth to a distant source of EM radiation. You can write 'space' and draw an arrow pointing to the right just below your square to label this axis.
7. Let R represent the current radius of the Big Bang's light cone, which is 14.43 billion light years. This establishes a distance scale on the horizontal axis of our diagram.
8. The diagonal BC describes a line on the Big Bang's light cone and represents the speed of light. It also represents an upper bound on the radius of the universe, since (except maybe for the inflationary epoch) the universe can't expand faster than the speed of light.
9. I am interested in establishing, on this space-time diagram, the trajectory of light from a distant source which arrives at Earth now.
10. The light from the distant source will arrive at E along the line ER on the diagram because it must arrive at the speed of light, and because of the way we have established the scales on our axes. The angle of incidence with EB must be 45 degrees.
11. Now, suppose that the light from the distant source narrowly missed an intermediate galaxy, G, say one that is now 2 billion light years from Earth. We would like to plot the location of galaxy G on our diagram at the position it was when the observed light from the distant source swept past it.
12. It might seem that G would fall somewhere on the line segment EM. After all, it has to come in at a 45 degree angle, and light goes as straight as anything we know.
13. For any point, G, on line segment EM, (except when G=E) the angle BGR is greater than 45 degrees. The speed of light is constant in each reference frame. BG forms the reference frame for galaxy G, so the path of the light from the distant source must form a 45 degree angle with BG. This is a contradiction which means that G cannot lie on line segment EM. It must lie somewhere below EM on our diagram.
14. The trajectory we seek must be a curve which intersects all lines containing B at a 45 degree angle. Such a curve is a logarithmic spiral. It is described by the equation r=3e^a in polar coordinates with the origin at B, and the angle, a, measured counterclockwise from BR. r is the distance from B. (I don't know how to display the Greek letters rho and theta on this forum, so I had to depart from the standard polar coordinate notation and use r and a instead.)
15. To plot this curve on our diagram, start by setting a=0. Then e^a=1 so r=3. If we scale our axes so that 1 unit = 1 billion light years on the horizontal axis, and 1 unit = 1 billion years on the vertical axis, then E and R each fall 14.43 units from B, and our curve begins at the point 3 units to the right of B. This is about a fifth of the way from B to E. I think it will be sufficient for you to just estimate the distance and mark the point. Label it P.
16. The curve intersects BR at P at a 45 degree angle going up to the right, parallel to BC.
17. The curve then bends to the left becoming vertical as it intersects BC. Let Q be the point of intersection. The curve continues to bend to the left until it passes through E at a 45 degree angle to line BE. This is because when a=pi/2, e^a=4.8 something, and so r=14.43.
18. We can calculate the location of Q by setting a=pi/4. Then r=3e^(pi/4)=6.57. The horizontal and vertical components of Q are each r/(sqrt(2)) = 4.6. This is a little less than a third of the way from B to C. You can now draw a short vertical line through BC at Q and with the end points at P and E, you can sketch in the spiral curve.
19. This curve seems to me to meet all the requirements for a plot on our diagram of the trajectory of distant EM radiation reaching Earth now. It also seems to me that Q represents the oldest, or most distant, possible source of EM radiation observable from Earth now.
20. The vertical distance from Q to BR is 4.6 billion years on our scale. That means that the oldest light we could possibly see would be about 9.8 billion years old. It would have originated when the universe was about 4.6 billion years old. Yet, I hear that scientists claim to observe CMB radiation that is a mere 300,000 years old (or at least they expect to make such observations from the Microwave Anisotropy Probe (MAP) satellite).
21. What have I done wrong?
Warm regards,
Paul |