3. All three solutions in 2.b correspond to de Sitter spacetime (or a portion of it), with different choices of coordinates. One way to define 4D de Sitter spacetime is via an embedding in 5D Minkoswki spacetime, 4 ds² = -(dxº)² + dx³ dx², i=1 a) Consider the coordinates with constraint l² = − (xº)² + Σx²¹x². a = V 2 – R? sinh(T/), x¹ = √²-R² cosh(T/l), 4 r = {sinh(t/l), x² = y¹l cosh(t/l). What is the constraint in terms of the coordinates y? Determine the induced line element on de Sitter spacetime. You should recover one of the cases in question 2. b) With the coordinates where j = 2, 3, 4, the analogous exercise (Assignment 4) gives -1 i=1 ds² (1 - R²³) ar² + (1 - 1²) dR² + R²³d$²₂) · 12 x³ = R₂³, These are known as static coordinates. The cosmological horizon is located at R = l. How long does it take for an observer at rest at R = 0 to receive a radially directed light ray emitted at R= Rel?
3. All three solutions in 2.b correspond to de Sitter spacetime (or a portion of it), with different choices of coordinates. One way to define 4D de Sitter spacetime is via an embedding in 5D Minkoswki spacetime, 4 ds² = -(dxº)² + dx³ dx², i=1 a) Consider the coordinates with constraint l² = − (xº)² + Σx²¹x². a = V 2 – R? sinh(T/), x¹ = √²-R² cosh(T/l), 4 r = {sinh(t/l), x² = y¹l cosh(t/l). What is the constraint in terms of the coordinates y? Determine the induced line element on de Sitter spacetime. You should recover one of the cases in question 2. b) With the coordinates where j = 2, 3, 4, the analogous exercise (Assignment 4) gives -1 i=1 ds² (1 - R²³) ar² + (1 - 1²) dR² + R²³d$²₂) · 12 x³ = R₂³, These are known as static coordinates. The cosmological horizon is located at R = l. How long does it take for an observer at rest at R = 0 to receive a radially directed light ray emitted at R= Rel?
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Part B - I don't understand why the last part of the line element has been excluded in the solution.
Transcribed Image Text:3. All three solutions in 2.b correspond to de Sitter spacetime (or a portion of it), with different
choices of coordinates. One way to define 4D de Sitter spacetime is via an embedding in 5D
Minkoswki spacetime,
4
ds² = −(dxº)² + dx³dx¹,
Σ
i=1
a) Consider the coordinates
xº
C2 _ R? sinh(T/),
where j = 2, 3, 4, the analogous exercise (Assignment 4) gives
−1
with constraint l² = − (xº)² + Σx²¹x².
x0 = {sinh(t/l), x² = y¹l cosh(t/l).
What is the constraint in terms of the coordinates y'? Determine the induced line element on de
Sitter spacetime. You should recover one of the cases in question 2.
b) With the coordinates
=
x¹
=
√12 R² cosh(T/l),
ds²
16² = -(1-1/²) α7² + (1-1/²)
dT²
4
1
i=1
dR² + R²d²(2).
x³ = R z³¹,
These are known as static coordinates. The cosmological horizon is located at R = l. How long
does it take for an observer at rest at R= 0 to receive a radially directed light ray emitted at
R = R₂ < l? Can a light ray be received from R =
=
=
Re > l?
Transcribed Image Text:17:35 Tue 3 Jan
b)
ds ² = 0 =2
El
d12²
(₁- (RIR) ²)~
(Notice that I is the proper time of the observer at R=0.)
hinand radially directed"
I arctank (Re)
от
=
AT
Re
d#²
V
qmplus.qmul.ac.uk
dR
1-RUP
Re
Real
Impossible to receive a
light may from Real.
3
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