Origins of the Expanding Universe: 1912-1932

Flagstaff, Arizona, USASeptember 13-15, 2012

Slipher‘s redshifts as support for de Sitter‘s universe?

Harry Nussbaumer Institute of AstronomyETH ZurichSwitzerland

Content

• The beginning of modern cosmology: 1917 Einstein and de Sitter

• De Sitter predicts redshifts. Do they explain Slipher‘s observations?

• Wirtz finds qualitative agreement between de Sitter’s theory and observation.

• Silberstein, Lundmark, Strömberg try to derive the ‘size’ of the universe. They all employ Slipher’s nebular wavelength-shifts.

• In 1925 Lemaître finds the catch in de Sitter‘s model.

He rediscovers dynamical solutions first found by Friedmann in 1922. In 1927 he develops the theory further and discovers the expanding universe. The rate of the drifting apart of spiral nebulae is calculated with Slipher‘s redshifts by Lemaître as well as by Hubble.

1917: The beginning of modern cosmology Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie published 1917

in «Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften».

In order to obtain a static universe Einstein introduces thecosmological constant λ, now mostly written as Λ .

Einstein‘s static model

The 4-dimensional universe projected into 2 dimensions.

The 3-dimensional homogenous, spatially closed part is projected into a circle. Its radius (radius of curvature) remains constant, there is no evolution.

De Sitter‘s empty universe (1917)

De Sitter studies the structure of an empty universe of constant radius of curvature:Time runs slower at large distances → redshifts

The greater the distance, the larger the redshift: open to observational verification.

Eddington 1923:The Mathematical Theory of Relativity

There is a second redshift effect: A particle at distance r is accelerated towards greater distances:

2

de Sitter: cos

1for .

2

rds R dt

R

rr R

R

2

2

1

3

d rr

ds

The greater the distance r, the larger the redshift Δλ

Eddington gives Δλ for 41 spiral nebulae, they are from Slipher

1924: Wirtz wants to check thevalidity of de Sitter’s universe.

42 nebulae in 6 groups with n members, apparent diameter D in arcminutes as a measure of distances. Redshifts from Slipher, apparent diameters from Curtis (Lick 1918) and Pease (Mt. Wilson 1919, 1920).

Log D v [km/s] n0.24 +827 90.43 +656 7 Wirtz finds a logarithmic relationship0.66 +512 8 v(km/s) = 2200 - 1200·log D. 0.88 +555 10 Because of the uncertainties in the1.07 +334 5 distances Wirtz considers this1.71 -20 3 relation as preliminary.

However, he draws the conclusion: There remains no doubt, velocities increase strongly with distance,as predicted by de Sitter.

Carl Wirtz (1876-1939)

1924: Remember the bone of contention of those years:

what is the Milky Way, what are spiral nebulae,

what is the universe?

In 1918 Shapley finds our place in the Milky Way, the size of which he vastly overestimates (Sun – galactic centre= 20‘000 pc).

He places the spiral nebulae at the outer confines of the Milky Way.

Others, among them Slipher, see spiral nebulae as galaxies (island universes).

However, in 1924 the issue is still open.

Harlow Shapley (1885-1972)

The size of the universe from de Sitter’s theory?

Carl Wirtz Spring of 1924 (submitted) They

Ludvik Silberstein, from January 1924 onwards) all get

Knut Lundmark August 1924 (submitted) line shifts

Gustaf Strömberg November 1924 (submitted) from Slipher

Silberstein’s wrong formula:

He derived it from de Sitter’s theory. But in his original 1924 publications he does not give his derivation. Eddington and later Lemaître critizise Silberstein’s formula as faulty: no negative sign! Silberstein employed it all the same on Cepheids, O-stars, globular clusters, spiral nebulae. – Choice of objects critizised by Lundmark.

From globular clusters and the Magellanic clouds he derived (as a mean from 4.4 to 9.1 •1012 a.u ): R= 6•1012 a.u.= 30 Mpc.

for d r

r RR

Lundmark: velocity-distance diagram(the first `Hubble diagram’)

Knut Lundmark (1889-1958)

The determination of the curvature of space-time inde Sitter‘s world. MNRAS 84, 747-764 (1924).

Lundmark investigated Cepheids, O-stars, globular clusters, spiral nebulae.His conclusion: the data do not reach sufficiently far out to allow cosmological conclusions.

Hubble 1929: de Sitter as an afterthought

Hubble investigates the solar motion:Determinations of the motion of the sun with respect to the extra-galactic nebulae ...

In 1928 he visits the IAU general assembly in Leiden and becomes aware that his data could be of value to the cosmological debate on de Sitter’s velocity-distance relation.

From his distances and Sliphers redshifts (gives no reference to the source of the redshifts) he finds: v ≈ Hr

Hubble’s conclusion: The outstanding feature, however, is the possibility that the velocity-distance relation may represent the de Sitter effect, and hence that numerical data may be introduced into discussions of the general curvature of space.

Hubble used Slipher’s redshifts, but there was one additional redshift from Humason (v = + 3779 km./sec. for N. G. C. 7619), which at the time was the largest known redshift. This one probably persuaded him to opt for a linear relationship (Sandage).

Einstein‘s field equations and cosmology

, 1, , 4

18

2ij ij ij ij

i j

R g R g GT

Einstein‘s field equations of 1915 link the geometry of spacetime to the energy-momentum tensor T. In 1917 he included the cosmological term Λ.From these equations the metric tensor gij, which gives the geometrical structure of the universe, can be derived.

22 2 2 2

22 2 2 2 2

for 0 Einstein

1cos for 0 deSitter

cos

drds R c dt ds c dt dr

R

dr rds R c dt ds c dt dr

rR RR

If we employ a spherical spatial coordinate system r, θ, φ and set θ=φ=0, then r is the proper radial distance and the line element for a light beam in the Einstein and de Sitter universes takes the form:

metric

In Einstein’s world the relation between dr and dt is the same everywhere.In de Sitter’s empty world, the relation depends on the distance from the observer.

Lemaitre: Why does de Sitter find redshifts?

2 2 2 2 2

2 1

1

( )

for 0 ( )

( ) ( )

( )v = H r

ds R t dr c dt

ds c dt R t dr

R t R t

R t

Lemaitre 1927: De Sitter violates the principle of homogeneity.

In his universe the world line is a geodesic only for the →

observer (A), but not for the observed (B, C).

Lemaître falls back on Einstein, but allows the radius of

curvature R to vary with time, R=R(t), as was already done by

Friedmann in 1922. His new insights lead to the discovery of

the expanding universe. →

In 1925 Georges Lemaître (1894-1966) spots the fallacy in de Sitters approach.

In 1927 has a fresh look at Einstein‘s fundamental equations and he discovers the

expanding universe.

Lemaître’s metric changes with time →

redshift if there is expansion

blueshift for contraction

Lemaître: redshifts and not blueshifts → the universe expands.

From averages he extracts the gradient for his relationship v= H·r.

→ H= 575 (km/s)/Mpc. The confirmation of linearity he leaves to the observers.

Lemaitre 1927 (Duerbeck)

He works with mean values of distances und velocities to obtain H= 575 (km/s)/Mpc

Hubble 1929

From mean values he obtains H= 530 (km/s)/Mpc.From best individual data he obtains H= 500 (km/s)/Mpc.

Reassuring observation for linearity:Humason gave him NGC 7619 withv= 3910 km/s and d≈ 7 Mpc

Conclusions:

Slipher’s redshifts were essential on two occasions during the discovery of the expanding

universe:

1. In 1917 they seemed to support de Sitter’s world and were thus the reason for

its extended discussion. During the following years they were an observational

cornerstone in the cosmological debate.

2. In 1927 they showed Lemaître that the universe was neither static, nor contracting,

but expanding. In addition they entered into the calculations of H by Lemaître in 1927

and Hubble in 1929.

Most of what is given in the presentation is contained in this book.