l l -waves and ads-plane waves in null aether...

Ç E T İ N Ş E N T Ü R K D E P A R T M E N T O F P H Y S I C S

B I L K E N T U N I V E R S I T Y

M A Y 2 4 - 2 7 , 2 0 1 6 I P M , I R A N

W O R K W I T H M . G Ü R S E S

( a r X i v : 1 6 0 4 . 0 2 2 6 6 )

𝑝𝑝-Waves and AdS-Plane Waves in Null Aether Theory

Outline

Null Aether Theory (NAT)

𝑝𝑝-Wave & Plane Wave Spacetimes

𝑝𝑝-Waves & Plane Waves in NAT

Kerr-Schild-Kundt (KSK) Class of Metrics

KSK Metrics in NAT

AdS-Plane Waves in NAT

Conclusion

Explicit Solution in D = 3 Special Solution in D > 3


NAT is described by

NAT

NAT is described by

𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔𝑐𝑜𝑛𝑠𝑡𝑠.

𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟

𝑎𝑒𝑡ℎ𝑒𝑟 𝑓𝑖𝑒𝑙𝑑

𝑐𝑜𝑠𝑚𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙𝑐𝑜𝑛𝑠𝑡.


NAT is described by

The aether field has the fixed-norm constraint

𝜀 = +1 ⟹ Einstein-Aether theory. [Jacobson & Mattingly (2001)]

introducedby λ

𝑠𝑖𝑔𝑛 = (−, +)

NAT

The eqns. of motion are

where and

From now on, 𝜀 = 0 ⟹ NAT.

𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛 − 𝐴𝑒𝑡ℎ𝑒𝑟𝑒𝑞𝑛𝑠.

𝐴𝑒𝑡ℎ𝑒𝑟𝑒𝑞𝑛.

𝑝𝑝-Wave Spacetimes

𝑝𝑝-waves (plane-fronted waves with parallel rays) are defined by

which immediately implies that

𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑡𝑙𝑦 𝑐𝑜𝑛𝑠𝑡. 𝑛𝑢𝑙𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 𝑓𝑖𝑒𝑙𝑑

𝑔𝑒𝑜𝑑𝑒𝑠𝑖𝑐 𝑤𝑖𝑡ℎ 𝑙𝜇 =𝑑𝑥𝜇

𝑑𝑣

𝑎𝑢𝑡𝑜𝑚𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑎 𝐾𝑖𝑙𝑙𝑖𝑛𝑔 𝑣𝑒𝑐𝑡𝑜𝑟

𝑙𝜇 = 𝜕𝜇𝑢

𝑝𝑝-Wave Spacetimes

Consider Kerr-Schild class of 𝑝𝑝-waves:

In the coord. sys. 𝑥𝜇 = (𝑢, 𝑣, 𝑥𝑖),

For such metrics,

𝑓𝑙𝑎𝑡𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑

𝑝𝑟𝑜𝑓𝑖𝑙𝑒 𝑓𝑢𝑛𝑐.

Plane Waves

Plane waves are subclass of 𝑝𝑝-waves for which

In Einstein gravity,

𝑖𝑛 𝐷 = 4,

𝑥𝑖 = (𝑥, 𝑦)

𝑠𝑦𝑚. 𝑚𝑎𝑡𝑟𝑖𝑥

𝑝𝑝-Waves in NAT

𝑝𝑝-Waves constitute exact solutions to NAT.

To show this, take the null aether field as

and assume that

so that

𝑠𝑐𝑎𝑙𝑎𝑟 𝑠𝑝𝑖𝑛−0 𝑎𝑒𝑡ℎ𝑒𝑟 𝑓𝑖𝑒𝑙𝑑


Then the field eqns. of NAT become

Klein-Gordon eqn.

⎕𝜙 − 𝑚2𝜙 = 0

𝑚2 ≡ −𝜆

𝑐1≥ 0

(𝑐1≠ 0)

null dust

𝑇𝜇𝜈 = ℇ𝑙𝜇𝑙𝜈

ℇ ≥ 0 ⇒ 𝑐3 ≤ 0 & 𝜆

𝑐1≤ 0


For 𝑝𝑝-waves spacetimes,

Thus, 𝑝𝑝-waves are solutions if the Laplace eqn. for 𝑉0, and the Klein-Gordon eqn. for 𝜙 are satisfied.

𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑎𝑛𝑠𝑎𝑡𝑧

𝑡ℎ𝑒𝑦 𝑑𝑒𝑐𝑜𝑢𝑝𝑙𝑒

Plane Waves in NAT

Plane waves can also be constructed:

when 𝑐3 = 0 ⟹ 𝑉 = 𝑉0:

when 𝑐3 ≠ 0, but 𝑉0 = 𝑡𝑖𝑗 𝑢 𝑥𝑖𝑥𝑗:

zero zero

(𝐷 = 4)


KSK class of metrics are defined by [Gürses, Şişman, Tekin (2012)]

⊕

𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑 𝐾𝑆 𝑓𝑜𝑟𝑚


KSK class of metrics are defined by [Gürses, Şişman, Tekin (2012)]

𝐾 > 0 𝐾 = 0 𝐾 < 0

𝑑𝑆𝑀

𝐴𝑑𝑆

⊕ 𝑚𝑎𝑥. 𝑠𝑦𝑚.𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑

𝐾𝑢𝑛𝑑𝑡 𝑐𝑙𝑎𝑠𝑠

𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑 𝐾𝑆 𝑓𝑜𝑟𝑚

KSK Metrics in NAT

Again assume that

but with

so that

𝑠𝑐𝑎𝑙𝑎𝑟 𝑠𝑝𝑖𝑛−0 𝑎𝑒𝑡ℎ𝑒𝑟 𝑓𝑖𝑒𝑙𝑑

Then the field eqns. of NAT become

For the KSK metrics, we have

Then we obtain

𝐹𝑜𝑟 𝐾 = 0 𝑎𝑛𝑑 𝜉𝜇 = 0,𝑤𝑒 𝑟𝑒𝑐𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑝𝑝−𝑤𝑎𝑣𝑒 𝑐𝑎𝑠𝑒!

Let us assume the ansatz

There are two possible choices for 𝛼:

Thus, exact wave solutions propagating in nonflat backgrounds can be constructed in NAT.

𝑓𝑜𝑟 𝛼 =𝑐3

2

𝑓𝑜𝑟 𝛼 =𝑐1 + 𝑐3

2

AdS-Plane Waves in NAT

Now assume that the background is AdS; i.e.,

Then taking , one can show that

𝑧 = 0 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠𝑡ℎ𝑒 𝐴𝑑𝑆 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦

Therefore, AdS-plane waves can be constructed as follows:

where and .

𝑓𝑜𝑟 𝛼 =𝑐3

2

𝑓𝑜𝑟 𝛼 =𝑐1 + 𝑐3

2

𝑜𝑟

AdS-Plane Waves in 3D

The field eqns. can be solved exactly in 3D because

The solution of the aether eqn. is

where

𝑓𝑜𝑟 𝑚 = 0

𝑓𝑜𝑟 𝑚 ≠ 0

𝑎1,2 𝑎𝑟𝑒 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦

𝑓𝑢𝑛𝑐𝑠.

And the Einstein-Aether eqns. become

when 𝑚ℓ ± 1 ≠ 0.

𝑐𝑎𝑛 𝑏𝑒 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑𝑖𝑛𝑡𝑜 𝐴𝑑𝑆


When 𝑚ℓ + 1 = 0,

When 𝑚ℓ − 1 = 0,


When 𝑚 = 0,


For a valid behavior as z → 0,

Thus, the AdS-plane wave solution is

AdS-Plane Waves in Higher D>3

In dimensions D>3, generic solution is not possible!

However, solutions can be obtained if we assume the wave is homogeneous along the transverse coords.:

In this case,

where

And

whose general solution is

On the other hand, when 𝑑+ = 0,

or, when 𝑑− = 0,

All these mean that

[𝐷 = 4 & Λ < 10−52 m−2 ≈ 10−84 GeV 2]

AdS-Plane Waves in Higher D>3

Then the solution is

𝑒𝑥𝑎𝑐𝑡 𝑝𝑙𝑎𝑛𝑒 𝑤𝑎𝑣𝑒 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑖𝑛𝑔𝑖𝑛 𝐷−𝑑𝑖𝑚. 𝐴𝑑𝑆 𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑

𝑖𝑛 𝑁𝐴𝑇

Conclusion

We constructed exact plane wave solutions in NAT.

These are important in that they are exact solutions.

Waves propagating in dS backgrounds can also be constructed! (In progress)

l l -waves and ads-plane waves in null aether...

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